I have done a LOT of classic PS1 games on my PSP and now my Vita the past few years and I think the following ones are some worth looking into:
Crash Bandicoot 1, 2 and 3 look absolutely incredible and hold up really well even today. Very simple platforming with minimal bullshit, some great levels and gameplay. I know many disagree with me but IMHO they hold up way better than Mario 64 does and are the absolute best retro 3D platforming games. I actually like Mario Galaxy way better than 64 because it feels it moved closer to Crash with focus on platforming. Great trilogy of classic games with minimal flaws but one that is worth mentioning is the terrible save system in Crash 1. You can only save when you complete a bonus round in a level or find a gem. 2 and 3 let you save whenever you want. Bonus is
Crash Team Racing Best kart racing game ever made imo. It does have decreased value as handheld game tho when it loses local multiplayer but if you are a nut for this stuff I would still recommend it as it holds up really well and the single player campaing is actually really fun with some good challenges if you want to do everything. I would recommend this for the PS3 more tho if you can access some multiplayer fun.
Final Fantasy 7, 8, 9 are still RPG classics. I especially would recommend 7 and 9. Prerendered backrounds help a lot with graphics holding up but 7 does have the terrible blocky characters that can be a little too much for some people that have no nostalgia glasses for them.
Metal Gear Solid is another great one and I am firmly in the camp that thinks it is better than Twin Snakes. Music is great and graphics hold up while in gameplay but it can look pretty jarring when they do closeups of characters in cutscenes. Story and characters are fun if you can sit through the sometimes long codec conversations.
Silent Hill is one of the games that hasn't exactly ages as well as some others especially when it comes to the graphics and teeeeeeerrrrible voice acting :d. However when it comes to horror there is no other game as old that holds up as well as this one. Athmosphere still holds up and when it comes to oppressive sounds and music this is the best game I have played in that regard. If you like horror games and can manage to sit trough bad graphics, VA and tank controls this is worth looking into since the things that this game does well it does REALLY well.
Resident Evil 1, 2 ,3 are some of my favourite games ever but I have this feeling I wouldn't recommend them so easily. Unlike Silent Hill these ones barely work as any horror anymore but what they do well is the gameplay itself. Tank controls are a big turnoff to some people but if you can look past that they are very well crafted survival games. 1 is the one that has aged the worst with SH levels of bad VA but it does have a newfound camp value and if you are old fan this one doesn't dissapoint even after being used to the vastly superior Gamecube Remake. Resident Evil 2 is the absolute gem of old retro RE and is the one I definately recommend the most. Best value for money when it comes to the series. VA is way better than original, graphics are more varied especially when it comes to the zombies and other monsters (original had like 3 different zombies while sequel has like, 10+?). Two different scenarios that actually makes 2 playthroughs more interesting than the original despite both having 2 playable characters. Resident Evil 3 is good also. Feels more modern than the other 2 and when it comes to gameplay it holds up the best. If you are new to the series I recommend checking out RE2 if you are interested for some retro survival games. Prerendered backrounds go a long way helping the graphics.
Tombi/Tomba holds up really well. Its hard to describe the game but it's kinda like if Zelda was a 2D platforming adventure game. Check it out!
Parasite Eve really interesting RPG and feels unique. Prerendered backrounds help with graphics yet again and the story is something different you are expected to see in most RPGs. Great lenght too and it keeps your interest and ends before you get bored of the thing, a curse I have for some RPG games nowdays.
Klonoa while it is very simple in design it has almost unbeatable charm when it comes to PS1 games. If you are fan of 2D platformers and want something light to play check this one out definately. It won't give you any challenge really but it is one of those games you enjoy from start to finish.
Rayman graphics look great and is just a great 2D platformer overall. If you don't like easy games like Klonoa go for this one. Rayman can be fuckin brutal when it feels like it and is actually one of the hardest PS1 games I have played. There are some life codes and stuff if you can't get through it legit. Best 2D platformer for the PS1 in my opinion when it comes to gameplay discussion only.
I also can't stress how great the music is!
On the other hand
Medievil is the kind of game I would recommend if you have nostalgia for it but othervise there are better games. I have fun with it but it is pretty dated game.
GAMES THAT DONT HOLD UP!:
Tomb Raider man I was surprised how badly this one has aged. If you want it for nostalgia reasons I would still recommend Tomb Raider Anniversary (has anyone played the PSP version? If that is decent I would actually be interested...). Just annoying game to play for me anymore and even nostalgia glasses couldn't save this one for me.
Gex Deep Cover Gecko This is like the anti Crash for me. I liked this one as a kid but I couldn't bother finishing it on PSP. Feels like a bad version of Mario 64 with platforming being really weak.