The ending of "Tails' New Home", where both Tails and Sonic come to accept that Sonic is Tails' family, after searching for his parents. Their dialogue... just read it yourself.
Sonic: "Sorry for all the trouble, squirt. I was just trying to find you a real family."
Tails: "I've got a real family. I've got the best!"
Sonic: "Oh yeah, where?"
Tails: "A family is just people who care about each other more than anyone else, right?"
Sonic: *gasps* "Right..."
Tails: "You're it! You're my mom, you're my dad and you're my picket fence!" *attempts to hug Sonic*
Sonic: "Woah! You're not gonna use the "L" word, are you?!"
Tails: *thoroughly embarassed* "Well..."
Sonic: *grabs Tails from behind and spins him over his head* "Then I will, because it's a great word. I love you, little bro! Now I've got something even more important to say: "Noogie time"!"
Tails: *laughing* "It's good to be home!"
In "Sonic The Matchmaker" Robotnik Jr. tries to win Breezy's heart by acting as a hero like Sonic, but fails at every attempt. At one point, it's Breezy herself (after Tails told her about Jr.'s feelings) who saves Jr. from a waterfall, and eventually realizes that she loves the feeling of being considered necessary in someone else's life. At the end of the episode, she proudly says to Sonic (her previous crush):
Breezy: "I'm sorry it can't work out between us, Sonic, but Jr. NEEDS me too much, and I really LIKE that!"
Which is immediately followed by another heartwarming moment, as Sonic responds "I know just what that means!" while putting an arm around Tails's shoulder. It might be considered a Ho Yay clue, but in actuality it shows how much having a little brother to protect, and receiving brotherly affection in exchange, makes Sonic happy and damned proud of it.
Another brotherly moment with Sonic and Tails happens near the ending of "Full Tilt Tails".
Tails: "Sonic, how come when I got what I always wanted more than anything else, everything went wrong?"
Sonic: "Sometimes you're just not ready for what you want the most. But I'll bet that you'll grow up and you'll be ready. You maybe even be as fast as me!"
Although it was bound to go south by the end of the episode, Robotnik's interaction with his robot son in "Robotnik, Jr." is quite wonderful to see. It's one of the most heartwarming moments in the show where Robotnik is truly happy.