Disney's Epic Mickey (if you're a Disney fan then it's a no-brainer. If not then it's still a great Action-Platformer)
Sonic Colors (The game that broke the Sonic Cycle. Give it a shot!)
Tatsunoko vs. Capcom (Despite not knowing the Tatsunoko crowd, they had some cool characters.)
Endless Ocean: Blue World (The sequel to the wonderful diving game. One of the most unique and relaxing games out there.)
No More Heroes/Desperate Struggle (The two excellent games from Suda51. Both are sure to make you a fan of Travis Touchdown.)
Silent Hill: Shattered Memories (A reimagining of the original Silent Hill, the game is unique, tense, and unforgettable.)
Muramasa: The Demon Blade (Another VanillaWare masterpiece. The art style is so incredible, and the gameplay is simple yet satisfying.)
Little King's Story (A very odd and quite wonderful RTS game. Pikmin-like... but also very much its own game. Quite hard to describe, but great!)
Okami (If you haven't played the PS2 version {I'm sure you have}, then you need to play this masterpiece).
Resident Evil 4: Wii Edition (I think this is the definitive version with it's IR controls.)

Even though they're 1st Party, don't miss out on...

Kirby's Epic Yarn (The graphics and music will make you smile, and the gameplay is very engaging!)
Donkey Kong Country Returns (Platforming brilliance by Retro Studios.)
Metroid Prime Trilogy (If you can get your hands on it... get your hands on it! The first two games are (imo) made even better by the IR controls).
Fire Emblem: Radiant Dawn (I love this series, and I think this is the best one. Try and get Path of Radiance on the GameCube too [it's the prequel])

Have fun Xen!

Silent Hill Shattered Memories and Virtual Console.

Thanks everyone, Smeags and bmmb especially! You mentioned lots of games I wanted, but managed to forget :D
And yes, I did play Okami on the PS2. Never got far (I didn't really try, it was a busy time) but I will get back to it soon enough.

Glad to see many already recommended Endless Ocean: Blue World (aka adventures of the deep) - mesmerizing game, that!

Also, even though it's technically a Nintendo game, it is nonetheless criminally under-appreciated - ExciteBots!
I myself am ashamed to only have gotten this game recently, it may look odd in videos but once you've played it you'll immediately see why it deserves all the praise it received - supports online play, too.

Another less known game that's worth a look is de Blob, or it's multi-plat sequel de Blob 2 which apparently controls a bit better.

A unique game with a surprising amount of polish for the Wii's early days is Capcom's Zack & Wiki, though it can be notoriously difficult and a bit unfair in the later levels, but still worth it IMO since it can be acquired cheaply.

Obviously there are the Nintendo Selects games that are all going for $19.99 such as PunchOut!!, WarioLand Shake, Mario Soccer, WarioWare Smooth Moves, Twilight Princess etc.

Speaking of cheap Nintendo games, Other M is definitely worth it for 20 bucks, and if you've never played the original, new play control DK Jungle Beat is a blast from the same team that went on to make the Mario Galaxy games. There's also npc Pikmin 1 and 2 if you've never played those.

Finally, a few more personal favorites of mine - Sky Crawlers (part anime, part flight combat, all fun) and Trauma Team (best game in the series). Both are also going for cheap!

Let us know what you end up getting, and enjoy your new Wii and its many games!

Oh yes, Sky Crawlers was (is!) on my radar for long. The general rule about me is... if it has anime, it'll at least get a look from me. If it has planes or mecha, it's a likely buy :D

I've also read an Excitebots review back in 2010 and it was pretty interesting. They said that it's kinda like the ultimate cartoon based game!

I'm guessing you already know about Sakura Wars, then. IGN gave it a 9!

Yep. Got it on PS2 more than a year ago though, it's the better overall package. I think the Wii version only has English VA (correct?). PS2 has both+2 artbooks+poster+better packaging :D

I love the PS2. Love it love it love it.

Sorry Rol, sex jokes and pretty 2D are both a guilty pleasure :P Those two are definite purchases.

Also, what's not to like about LKS?

I assume you know of the awsesomeness of DKCR. Aside from that, definitly check out Kirby's Return to Dreamland, Little King's Story, Sonic Colors, Dawn of Discovery, Zack and Wiki, Monster Hunter Tri Ivy the Kiwi, Kirby's Epic Yarn, Red Steel 2, Madworld.. Hmm that's about all I got off the top of my head, at least in terms of the not-so-obvious must haves or close to em..

Donkey Kong Country Returns
Monster Hunter Tri
World Of Goo
Cave Story
Wii Party