Not best games for learning necessarily, just in general must play games. I honestly don't know too well. I just picked up Megami Ibunroku Devil Survivor. But I want to know some other cool games in general. What are your favorites, or what do you know is popular for sure, I am finally at the level to enjoy most of these games so I'd like to get started. Please share!

tenchu 3/tenchu kurenai/tenhu 4 are all for psp now. I recommend those, those are fun to play. Here's the official site. I.e ninja games, all about stealth/killing. The story is simple good vs evil type thing in feudal japan time-era.

http://www.tenchu.net/

I'm partial to Taiko No Tatsujin, a rhythm game that's available for both PSP and DS.

There are the obvious zelda DS games, plus the Mario & Luigi RPG games who are all quite fun to play. I'm also a fan of the fire emblem series , or basically anything Intelligent Systems makes...

DS:
- 逆転裁判 series (phoenix wright etc), though I'd recommend playing them in english for enjoyment since the localization team did quite a good job at including pop culture references and adapting wordplays and stuff.
- 押忍！闘え！応援団 crazy fun rhythm game imho, I actually got a ds just to play this. No japanese knowledge required to play tho.
- Prof. Layton series, basically a point and click adventure featuring hundreds of puzzles of varying difficulty.
- すばらしきこのせかい (the world ends with you) a rpg-ish game with controls that will drive you crazy, but totally worth it imho.
There's plenty of rpg games actually, I don't even know where to start from... The always popular final fantasy (remakes mostly), zelda, dragon quest... those didn't really impress me though. There's a Chrono Trigger DS iirc, which might be worth it if you are nostalgic of the snes era. Tales of hearts looked pretty good at the time, it also has spoken dialogues which is always a plus, same for suikoden tierkreis. Haven't really played them but kingdom hearts and blood of bahamut are supposedly good games too. If you can't seem to find an rpg that suits your tastes you can always make your own with rpg-maker ds too =)
Going for the more obscure stuff:
- 7th dragon (the spiritual sequel to etryan odyssey) is a pretty interesting dungeon crawler, hard battles and interesting party mechanics
- Estpolis DS, DS installment of the Estpolis (Lufia in us/eu) snes series, worth it only if you played the snes installments basically, the game is mediocre but remakes of the original ost make up for it.
- 極限脱出 9時間9人9の扉 a graphic adventure game, seems interesting but haven't really looked much into it.
Also on my "to play sooner or later list" (just throwing titles): Twilight syndrome (survival horror?), Item getter, Hayarigami 2

PSP:
- 428 ～封鎖された渋谷で basically a visual novel, but really badass at visual novelling =p scored 40/40 on famitsu which is probably a first for this kind of game. Japanese level is a bit too high for me though, I'm at 1-2k vocab past core 6k and I need to check 1 word per sentence on average, which is really slowing me down (though I can get the gist of it without a dictionary for the most part)
- Persona3 portable (assuming you haven't played it on ps2 yet), rpg with some dating sim elements, demons and stuff. You probably have heard of/played it already if got yourself Devil Survivor for the DS i guess.
Final fantasy crisis core and final fantasy dissidia are quite popular, haven't really tryed them out yet though. Um... and disgaea, I'm pretty sure psp have it (possibly ds too). Lot's of "tales of" installments too. Dunno haven't really played much aside from the first two I listed on my psp.

I had just recommended 428 in another thread, but it looks like Seizarさん beat me to the punch in this thread! Or maybe I did - we basically posted at the exact same time Awesome.

For DS, I highly recommend the 世界樹の迷宮 games (also known as Etrian Odyssey outside of Japan) if you're into dungeon crawlers. The gameplay is actually quite hardcore, basically Wizardry-style dungeon exploration with loads of customization and steep difficulty curves. The unique thing about this series is that you're forced to draw your own dungeon maps with the stylus! It sounds daunting at first, but it's actually one of the most interesting and intuitive parts of the game. Definitely not a series for everyone, but if you enjoy dungeon crawlers, don't mind a bit of grinding and the concept of drawing your own maps, you'll probably love em.

Kind of a kids game
1010 - San-X Chara Sagashi Land

but very nostalgic and fun. Its a spot the difference type of game. Strangely addictive and especially fun with your gf ^^

Thanks, those will definitely get me started!, Any more?