If you're looking for peoples' personal console favorites, I've got a big list here that you can either emulate on PSP or PC (I'll list which). I'll also try to separate by series and genre. When I suggest a whole series, I am suggesting you try the first one and play the rest if you like it, obsiously.
Actually PSP, or better emulated on PSP:
RPGs:
Pokémon Fire Red/Leaf Green (GBA) - this applies to all main series Pokémon games on GB/GBC/GBA, these are just the ones I feel may be best suited for a beginner to the series
Megaman Battle Network 1-6 (GBA)
Tales of Phantasia (PSX) - with an English translation patch you can easily find, and you will need to look up how to convert PSX games to PSP
Tales of Destiny (PSX) - released in English, though nowhere near as good as Phantasia
Tales of Eternia (PSP)
Final Fantasy VII (PSX) - I bet you saw this one coming
Crisis Core: Final Fantasy VII (PSP)
Kingdom Hearts: Birth by Sleep (PSP) - be sure to play this series in release order, the story is way too convoluted to juggle games
Platformers:
Sonic the Hedgehog 1-3 and Sonic & Knuckles (Mega Drive/Genesis)
Sonic Advance 1-3 (GBA)
Super Mario Bros. and World series (NES/SNES) - some are remade in the Super Mario Advance series, see below
Super Mario Advance 1-4 (GBA) - in this case, I especially suggest 4
Megaman 1-6 (NES)
Megaman X1-X3 (SNES)
Megaman Powered Up (PSP) - remake of Megaman (NES)
Megaman Maverick Hunter X (PSP) - remake of Megaman X (SNES), see below
Megaman Zero 1-4 (GBA)
Adventure:
The Legend of Zelda (NES)
The Legend of Zelda: A Link to the Past (SNES/GBA)
Emulated on PC:
RPGs:
Kingdom Hearts 1, Re: Chain of Memories and 2 (PS2) - in this order, with Birth by Sleep afterwards
Tales of the Abyss (PS2)
Tales of Sypmhonia (GameCube)
Platformers:
Sonic Adventure 2: Battle (GameCube)
Megaman X7-X8 (PS2)
Adventure:
The Legend of Zelda: Ocarina of Time (N64)
Trade-off between portability and performance or button accuracy:
Platformers:
Megaman X4-X6 (PSX) - They run perfectly on PSP, since you can convert PSX games, but having to map L2/R2 to either the analog stick or the D-Pad may be an issue with platformers, which is why I suggested the PSX RPGs in the PSP section, but these here
This is a long, but not all-inclusive list of my personal favorites. They are too many to remember. I'll be sure to write more of them if they come to mind.