For Flash Memory, any SD card will work (in teh SD slots of course). For Gamecube games you still need Gamecube memory cards (slot is on top with plugs for Gamecube controllers). I don't think the Wii can see or let you see on screen a USB flash drive.
Wii documentation does state to only use Wii branded USB wired LAN adapters, but since it has built in Wi-Fi I would just use that instead of running cable.
I just got one in my house for christmas, so take what I say with a grain of salt.