get a wireless G router instead of a lan adapter since it will serve you better in the long run, find one on discount if you can, would be cheaper than 30 bucks.

cube has a lot of good games, hard to get to them myself since I'm too busy playing Wii games.

I'd tell you to buy both Metroid Primes for GCN, but I'd just wait for the NPC! versions later on this year, if the presumably marginally higher cost doesn't turn you off.

wireless router all the way

I have one of these:

It works for both Wii & DS   :)

that requires your computer to be on which sucks more power than a router, if electricity bill is not an issue though then it's ok.

Lan adapter, buy Wind Waker, Twilight Princess, Eternal Darkness, I would say Pikmin and MP but you can get those for Wii.

i implore to try both the pikmin games. either the original gamecube versions or the wiimakes with wiimote controls added will do nicely.

I need to get some Wii internet myself,but I do not know how.

I had to buy the one from Nintendo because it was the only one that worked when the Wii launched, I'm not sure if other brands work now.

GCN games:

F-Zero GX
Beach Spikers (fun 4 player volleyball game)
Battalion Wars
Super Monkey Ball 1&2
Mario Golf: Toadstool Tour
Time Splitters 2
Luigi's Mansion
Killer 7
Ikaruga (you have to get this if you are into flying shooters)
Mario Kart: Double Dash
Super Smash Bros Melee
Worms 3d (I enjoyed it even though I probably prefer 2d Worms more)
Legend of Zelda: Wind Waker