Call its theme music 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 One 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.
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.
"The Forge." Captain Archer and Ambassador 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.
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 Lt. Malcolm 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 Sato. 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:
Malcolm: Pineapple?! That's my favourite... How on Earth did you know?
Hoshi: (sharing a grin with Archer) We have our sources.
It's a borderline-villainous example that's best not thought too much about, but in the Mirrorverse 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 T'Pol asked him not to tell anyone, and that he actually respected her privacy. Given the nature of the Mirrorverse 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.
T'Pol: I'm surprised you're not exhausted from all the beds you've jumped into recently. Hoshi: Commander Tucker told me you could use a few pointers in that area.
"Twilight" is widely considered one of the best (if not the best) episodes of Enterprise. 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.
T'Pol trying to comfort Archer's dog Porthos after the captain dies 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.
Ambassador 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.
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 sad goodbye.
In the episode 'The Catwalk', the 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 All right. Twenty square meters.
The little smile she gives him is what really sells it.
Archer's farewell speech in "Azati Prime:"
"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."
This page wouldn't be complete without mentioning Archer's "Lets explore it together" speech from "Terra Prime." Pitch perfect example of this trope as well as Rousing Speech and capped by Soval leading a Slow Clap.
"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.
"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."
Also, the scene between Archer and T'pol in "These are the Voyages" right before he gives his address at the founding of the Coalition of Planets.
Whatever feelings you may have towards the series finale, you gotta love the final lines from "These are the Voyages".
Captain Jean-Luc Picard: Space, the final frontier. These are the voyages of the Starship Enterprise. Its continuing mission...
Captain James T. Kirk: ...to explore strange new worlds, to seek out new life, and new civilizations...
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.
In "The Expanse," Soval completely opposes the Delphic Expanse mission and recalls T'Pol. After thinking it through, T'Pol resigns her commission. Archer is stunned.
"You need me."
Archer's speech in "Home":
"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."
Reed risking his own life to try and save Trip's life in "United". For that matter, just about any time Malcolm risks his own life to save another crew member. 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.
While it's only briefly mentioned, and rather offhandedly. It's pretty awesome that the Klingon Empire will become a member of the United Federation of planets 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.
In a way 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.
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 C.O. 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 "Horizon," Travis learns that his family's cargo ship, Horizon, will be passing near them and decides to visit. Unfortunately, he learns that his father has died several weeks ago. Travis tells Archer how his father had been very disappointed when he joined Starfleet, as he had been groomed to take command of Horizon. Archer, however, comforts him by recalling the recommendation letter from Mr. Mayweather when he was looking at candidates for helmsman. 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.
The end of "Storm Front, Part II." 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.