Archer encouraging Hoshi to step up and talk to the alien captain while Enterprise is under attack.
Hoshi: You don't understand. I don't even know how to say "pump". I'll get it all wrong, it'll just make things worse.
Archer: Things can't get much worse. [...] Hoshi, Hoshi, I need you to do this. We all do. That's why you're here.
T'Pol even gets in on the whole encouragement bit, telling Hoshi her words don't have to be perfect and even addressing her by her first name.
Although it's a small moment, "Silent Enemy" has one. In the middle of fighting off some rather nasty aliens, Archer makes it a priority to find out Reed's favorite food to make him a birthday dinner/cake. Reed, as it turns out, doesn't express preferences to anyone about anything, necessitating some clever detective work from Hoshi. Turns out Reed takes injections to control his allergy to bromelin, found in pineapple. Culminates in this cute exchange, as Malcolm cuts open the cake:
Reed: Pineapple?! That's my favourite... How on Earth did you know?
Hoshi: (sharing a grin with Archer) We have our sources.
T'Pol in "Fortunate Son" helping the little girl Nadine during her game of hide and seek. It's surprisingly adorable, yet totally in character for a Vulcan.
"Sleeping Dogs" has a lovely Friendship Moment where T'Pol comforts a panicky Hoshi and shows her a meditation technique to calm herself down.
"Shadows of P'Jem" has a nice little moment. When Archer finds out T'Pol is being transferred in retaliation for the P'Jem incident, he decides to take her along on a first contact mission instead of Trip as planned. Archer says he thought she might like one final mission with her captain. When she doesn't respond, he suggests he could just take her back if she wants. After a moment, she says, "We're only a few minutes from the capital. It would be a waste of fuel."
Likewise, at the end of the episode, after Archer has convinced Sopek to speak to the Vulcan High Command about rescinding the transfer:
T'Pol: You should have consulted me first. Archer: It's probably not too late, if you want to catch up to him. T'Pol: [beat] That won't be possible. Leaving sickbay would violate my doctor's orders.
Trip encouraging Kov to re-connect with his ailing father in "Fusion".
From the same episode, Archer checking on T'Pol at the end after her ordeal with Tolaris. They may not agree on everything, but it's clear the captain cares for his Vulcan chaperon.
At the end of "Shockwave", T'Pol stands up to Soval on behalf of the crew, even going so far as to point out that the destruction of the monastery on P'Jem (which he blamed Archer for) was the Vulcans' fault.
In "Minefield," while trying to defuse the mine, Archer and Reed begin discussing the former's style of command.
Reed: Frankly, sir, from my point of view, that kind of socializing has no place on a starship. Archer: I had a CO once, felt the same way. "They're your crew, not your friends." I thought about that a lot...when I took this command, but then I realized this is not a typical mission. We could be out here for years. All we have to depend on...is each other.
In "The Seventh", T'Pol is given a secret mission to apprehend a Vulcan fugitive, and after initially keeping Archer in the dark, she requests he come with her. When he presses her on why, her answer is, "I need to be with someone I can trust."
In "The Catwalk," Enterprise runs into a severe radiation storm; as a result, the entire crew has to evacuate into the nacelles, which are the only part of the ship heavily-shielded enough to keep them alive. During the hurried preparations, a heartbroken Phlox realizes that T'Pol has only allocated him fifteen cubic meters for use as an emergency sickbay. Why is this a problem? It's only enough room for two-thirds of his pets' cages.
Phlox: I'm not accustomed to making emotional appeals. Please don't make me choose between them.
T'Pol: (pause) All right. Twenty square meters.
The little smile she gives him is what really sells it.
In "Horizon," Travis learns that his family's cargo ship, the ECS Horizon, will be passing near them and decides to visit. Unfortunately, he learns that his father died several weeks ago. Travis later tells Archer how his father had been very disappointed when he joined Starfleet, as he had been groomed to take command of the Horizon. Archer, however, comforts his helmsman by recalling the recommendation letter from Mr. Mayweather when he was looking at candidates. All it said was this: "I've never met a more natural stick-and-rudder man, and you'll be a fool to not pick my son." For a character who was essentially an Advertised Extra, it's an extremely effective moment for Travis.
In "First Flight," while searching for a dark matter nebula, Archer tells T'Pol about his recently-deceased friend A.G. Robinson, who helped him save the NX Program years ago. After they successfully find and illuminate the nebula, Archer suggests calling it the "T'Pol-Archer Nebula," but T'Pol offers a suggestion of her own: the "Robinson Nebula."
"Twilight" is widely considered one of the best (if not the best) episodes of Enterprise, and it's loaded with heartwarming moments.
T'Pol's devotion to Archer is such that she resigns as Enterprise's captain to care for him. Every few hours, he forgets everything that happened after he was infected by the brain parasites, and she's always there to catch him up. Phlox expresses the opinion that she's in love with Archer.
In the climax, Archer attempts to create a subspace implosion to try and reset history and avoid the extinction of humanity. His ship is being blasted to pieces under his feet, his crew is being gunned down left and right by Xindi boarding parties, and the Xindi are on the cusp of completing the final massacre of the remaining humans. All of this, and the former captain still finds time to tell T'Pol and Phlox to try and escape the ship as the Xindi are only after humans and if this fails, they may yet survive. And the truly heartwarming part, of course, neither of the aliens so much as budges! Phlox merely looks at Archer, and T'Pol actually joins him in programming the warp core to implode. Archer notices them staying and angrily states "That's an order!" T'Pol almost casually reminds him "With all due respect, you were relieved of duty."
Even sweeter is the moment just after that when T'Pol gets shot and Archer stops what he's doing to catch her and lower her gently to the ground.
Archer's farewell speech in "Azati Prime".
Archer: I'm going to ask all of you to think back to the day when this ship was first launched. We were explorers then. When all this is over, when Earth is safe, I want you to get back to that job. There are four hundred billion stars in our galaxy; we've only explored a tiny fraction. You have a lot of work to do. Of all the captains who will sit in this chair, I can't imagine any of them being more proud than I am right now.
While it's only briefly mentioned, and rather offhandedly, it's pretty awesome that the Klingon Empire will become a member of the Federation by the 26th century. It also confirms the Organians' theory that humans and Klingons would become true friends. Nice to see the whole thing come full circle.
It also helps in that, in a way, it eliminates the fears and worries of the Klingon scientists and lawyers about the warrior class taking over the Empire.
The same episode also suggests that the Xindi have joined the Federation, both fulfilling Archer's hopes for peace with them and allowing them to help fight off the trans-dimensional aliens who spent a century manipulating them.
T'Pol trying to comfort Porthos after Archer is believed to have died stopping the Xindi superweapon. Heartwarming all on its own, but even moreso when you think back to her initial reaction upon meeting the captain's pet.
The end of "Storm Front, Part 2." After a year of fighting their way through the Expanse to stop the Xindi from destroying Earth, after the "WTF?" ending of "Zero Hour," and after two episodes of fighting alien Nazis (again, "WTF?"), it's nice to see our battered and exhausted heroes finally return home.
Archer's speech in "Home."
Archer: I've been told that people are calling us heroes. When it comes to my crew, you won't get any argument from me. But I think it's important that we remember the heroes who aren't with us: the twenty-seven crewmen who didn't make it back. Without their sacrifice, I wouldn't be standing here right now; none of us would. But I'm sure I speak on behalf of my entire crew when I say...it's good to be home.
From the same episode, Soval actually thanking Archer for completing his mission in the Delphic Expanse and going one step further by stating that although he initially disapproved of Archer being chosen as captain of Enterprise, he's clearly the right man for the job.
Archer: Ambassador. I want to apologize for what I said to you the other day. I was out of line.
Soval: Yes, you were.
Archer: I'm glad we can finally agree on something.
Soval: Captain...your actions, while being morally questionable, were necessary. I opposed your appointment as captain of Enterprise, but it's obvious now that I was wrong. T'Pol has told me that the Expanse would have someday encompassed hundreds of systems, including Vulcan. You've done a great service for both our worlds. (holds out his hand) Thank you.
(They shake hands)
In "Borderland" Enterprise prepares to leave Earth orbit on their mission to find the augments. Archer takes a look around the bridge at the crew, the warmth and pride evident in his expression.
Archer: Here we are again. Wouldn't have it any other way.
In Season 4's "The Forge," Archer and Soval, who've never seen eye-to-eye at any point in the series, stand over the coffin of Admiral Forrest, who gave his life to save Soval. Soval, with his newfound respect for Forrest, comforts Archer on the loss of his friend. This marks a turning point in the relationship between Humans and Vulcans.
Soval: His death is a loss to both our worlds. [...] He saved my life in the explosion. He could have saved himself.
Archer: He always put the mission first.
Soval: The last time we spoke, he was anticipating the prospect of joint missions, Humans and Vulcans working together.
Archer: We have to make that possible, despite the High Command.
Soval: It makes no sense to think the Syrrannites are responsible for this outrage.
Archer: We have DNA evidence.
Soval: Re-check everything, Captain, question everything, and don't let them keep you on Enterprise. The answers you need are on Vulcan, and however far you have to take this investigation, you have my support.
In "Observer Effect," Archer risking his life by exposing himself to the virus in an attempt to save Hoshi.
Also, just after that, when Trip temporarily dies, Archer rests a hand on his chest in a really tender and sadgoodbye.
Reed risking his own life to try and save Trip's life in "United."
For that matter, just about any time Reed risks his own life to save another crewmember. He might not be very good at expressing it in words, but it's obvious through his actions that he's come to care a lot for his friends on Enterprise, despite being pretty distant at first.
It's a borderline-villainous example that's best not thought too much about, but in the Mirror Universe, T'Pol asked Trip to help her through her pon farr. While his "help" wasn't exactly motivated by altruism, in a blink-and-you'll miss-it exchange, it's revealed that Mirror!T'Pol asked him not to tell anyone, and that he actually respected her privacy. Given the nature of the Mirror Universe, you would expect Mirror!Trip to have been bragging about banging the Vulcan. The fact that he didn't says a lot about his character.
Until a line from Hoshi suggests that he did tell.
Mirror!T'Pol: I'm surprised you're not exhausted from all the beds you've jumped into recently. Mirror!Hoshi: Commander Tucker told me you could use a few pointers in that area.
This page wouldn't be complete without mentioning Archer's speech from "Terra Prime." Pitch-perfect example of this trope as well as Rousing Speech and capped by Soval leading a Slow Clap.
Archer: Up until about a hundred years ago, there was one question that burned in every human, that made us study the stars and dream of traveling to them—"Are we alone?" Our generation is privileged to know the answer to that question. We are all explorers driven to know what's over the horizon, what's beyond our own shores. And yet the more I've experienced, the more I've learned that no matter how far we travel, or how fast we get there, the most profound discoveries are not necessarily beyond that next star. They're within us, woven into the threads that bind us, all of us, to each other. A final frontier begins in this hall. Let's explore it together.
Especially heartwarming considering that when the show started, they hated each other. Now they're Fire-Forged Friends.
There's also Phlox's admission to Archer, as baby Elizabeth's condition worsens.
Phlox: When you invited me to join this crew, I thought it would be an interesting...diversion for a few months; some time away from the complications of family which, on Denobula, can be extremely complicated. I didn't expect to gain another family. It hurts as if she were my child.
In the same episode, the characters land at Carl Sagan Memorial Station on Mars. The memorial at the station carries a particularly good quote of Sagan's.
"Whatever the reason you're on Mars, I'm glad you're there, and I wish I was with you."
The scene between Archer and T'Pol in "These Are the Voyages..." right before he gives his address at the founding of the Federation.
Whatever feelings you may have towards the series finale, you gotta love the final lines:
Captain Jean-Luc Picard: Space, the final frontier. These are the voyages of the Starship Enterprise. Its continuing mission...
Captain Jonathan Archer: ...to boldly go where no man has gone before.
A scene we almost didn't get. It wasn't in the initial draft of the script, but the writers fought for it and got it in.
Call its theme song cheesy and/or contrary to Star Trek tradition if you want, but the show's opening sequence depicting humanity's progress from sea travel to flight to space exploration had a very heartwarming hope-for-the-future impact on Season 1 viewers, particularly in contrast to the 9/11 gloom that dominated the media at the time. The more so, in that all but the last three innovations shown (a space orbiter, the Phoenix, and the NX-01) already existed in Real Life.
And, honestly, the theme of the song fits very well. At the beginning, Earth's crawling out of its worst war yet, Andoria and Vulcan are at each other's throats, and Tellar Prime is just out against everybody. And yet, by the end, these four planets will come together and form the Federation. Together, they will explore strange new worlds. They will seek out new life and new civilizations. They will boldly go where no one has gone before. Because it's out there and they can.