Plenty of these in the series, but PJ confessing his love to Rose and Max choosing to face a blizzard to find Goofy during the Christmas episode are some of the most memorable ones.
The chestnut speech in "O, R-V, I N-V U". Especially if you identify with PJ. And if you keep in mind that Goofy is effectively serving as a Parental Substitute for PJ's own (inadequate) father. What really makes the scene is how profoundly the speech affected PJ, who changed from dismissing the chestnuts to gathering them in large quantities, which even turned out to be a Chekhov's Gun. Then when he does "come out of his shell" in the second movie... it becomes Heartwarming in Hindsight too.
Goofy: Is there something wrong?
PJ(plaintively): Do I look dumb to you?
Goofy: No way, PJ! Here's dumb! (tosses chestnut at PJ's head)
PJ: Ow! (looks at nut) What's this?
Goofy: A chestnut.
Goofy: Not too perty on the outside, huh? It's all spiny and dull-looking.
Goofy: (takes chestnut from PJ and begins shelling it) But come the right time, the shell falls off and walla! Look at it! Ah-hyuck! (inner nut is shown) It's chock-full of good stuff, (cuddles PJ) and handsome too. Ah-hyuck! (touches PJ's nose affectionately) You know what I mean?
PJ(smiling softly): Yeah, I think I do...
Max coming up with two very elaborate plans to make it so that PJ could be friends with him in "Good Neighbor Goof" (the second half of the pilot), after PJ had been abused in Max's earshot, had been forbidden from seeing him, had revealed that Max was a Living Emotional Crutch, and had given up. It really seems to excuse any moment Max seems to take PJ for granted, and completely justifies PJ's fervently loyal behavior to Max in later episodes.
Max (and Peg) defending Goofy from Ms. Pennypacker in "Date with Destiny", Goofy rushing over to comfort Max after his nightmare, and Ms. Pennypacker taking back her judgment when she sees Goofy preparing Max for his second night with the Petes.
Goofy and Max winning the best father and son pair in the class in "Meanwhile, Back at the Ramp" because they'd do anything for each other.
Even Pete of all people gets a few, like the time he gave a Participation Trophy to the kid at the end of "Tee for Two".
Max figuring out which one of Goofy and the doppelganger was his real father in "Counterfeit Goof" after Goofy stepped in the wallpaper paste, and showing it by hugging him.
The Max and PJ subplot in "Everything's Coming Up Goofy" is really sweet. They are cautious about each other at first, but Max becomes very excited once he sees all the cool toys in PJ's room... until he finds out that Pete doesn't let him play with any of them. He shows immediate sympathy for his despondent new neighbor, and then finds a loophole in the rules that allows PJ to play with his toy tank. After everyone goes to bed, we see a shot of PJ, cuddling the tank like a teddy bear, and smiling in his sleep... one of the only genuine smiles we've yet to see him give. Aww.
Witnessing mobsters at the docks advance on the naive Goofy, the unseen Pete doesn't hightail it out of there despite previously warning him not to approach them to the point of exasperation. Rather, his priority was getting Goofy out of there instead of saving himself.