By far the best and most emotional episode is probably “Pinkie Pride”, one of the few truly musical episodes from a cartoon full of singing. Fully half the episode is songs, and it even guest stars Weird Al. One of the songs especially, Pinkie’s Lament, is a sad, slow song contrary to the many many happy and upbeat song in the series by that character. Its also the best song in the series. However, without a background in how Pinkie Pie typically is and perceived by everyone, you may not get the emotional jolt many people did when they watched this.
Try first “A Friend in Deed”, also a Pinkie episode with a great song, to see what kind of Pony she is.
For a good Fluttershy episode, there’s “Dragonshy” or “Hurricane Fluttershy”. She’s a very shy and demure pony that sometimes you just want to grab and shake to snap her out of it, but when she comes through for her friends its hilarious to see the change in personality
A more ensemble episode with a great story and song is the 2 parter the ends season 2 named “A Canterlot Wedding” part 1 and part 2. This is a great episode for the cast and especially for the technically main character Twilight Sparkle.
For a Rainbow Dash episode, try “Sonic Rainboom” and immediately follow that up with “The Cutie Mark Chronicles”. Rainbow’s the boyish jock of the group, always wanting to compete and showoff. The first episode is her own solo episode and the 2nd one is more of an ensemble one that gives you a bit of history on all the other characters.
Applejack is the country girl and her episodes generally deal with her farm that she works on with her grandmother, brother, and little sister. “Super Speedy Cider Squeezy 6000” is a hilarious episode with a great catchy song that took its inspiration from the Simpsons’ Monorail episode and song.
“Luna Eclipsed” features one of the most popular supporting characters, the formerly evil sister of the pony realm ruler. This episode is just downright hilarious. Lots of gags and memorable quotes.
Balance named a pretty good Rarity episode in “Suited for Success”. I would also add that “Simple Ways” is also really good, though she co-stars with Applejack in that one. It gets her personality through pretty well though
If you don’t have time to watch all of them, you can try watching some of the songs and see if you think they are catchy enough to find out the context. Some of the best songs in the series are “Winter Wrap Up”, “Find a Pet Song”, “The Flim Flam Brothers song”, “Smile song” (warning, do not click on Smile HD), “This Day Aria”, “Bats”, “Apples to the Core” and the aforementioned all songs from Pinkie Pride