Super Smash Bros. Brawl. Obviously.
The Trauma Center games are awesome. Surgery is more fun than it ought to be. Turbulence, dark tunnels, time bombs, toy locks, manmade pathogens, the scenarios and story are certainly NOT what you would see on an episode of ER.
Super Paper Mario is awesome. It's like oldschool platforming with a neato "3D rotation" twist and the humour and RPG elements taken from the Paper Mario RPGs.
Look up Zack & Wiki on GameTrailers.com. It's another one of those underappreciated games. Think point-and-click adventure with Wii Remote motion controls when using items. It's like Eternal Darkness, Beyond Good & Evil, and Psychonauts in that people won't stop saying how awesome it is so in some circles you might hear too much about it yet some groups of people have no idea.
(In the same vein, they say Okami's like that. That game with the dog and the inkbrush to do attacks or spells or something, now with Wii controls? I have yet to try this out myself, though.
I also haven't played No More Heroes, but it's developed by the same guy who did Killer7 (GameCube), one of the most uniquely demented games out there, that I want to make time for it. Since you'll be able to play GameCube games, try Killer7 one day, too.)
In general anything published/developed by Nintendo is good, though I personally have not yet tried Excite Truck, Wii Play, or Mario Kart Wii yet. Ask me about anything else by Nintendo, and I've likely played it (and loved it).
Get the Wii connected to the Internet. I had mine connected with the official wireless USB connector ('cause I had it before for my DS) but personally I find the wired connection better (using the LAN Adapter that you stick in one of the back USB ports). Either way you have to buy a part from Nintendo, you can't just stick the ethernet cable straight into the Wii. >_<
Then you can purchase WiiWare games, which are new games like Final Fantasy Crystal Chronicles: My Life As A King, which my brother absolutely loves. (It's like a city management sim where you send out parties of adventurers to dungeons to play Final Fantasy for you.) It's surprisingly deep and good-looking for a downloadable game, it looks like it could've sold well even if put on a disc. Takes up a lot of room on your Wii, but if you don't plan on getting any VC games, then you'll be more than okay.