Topic: Can someone explain to me how to download games on the Wii E Shop?

If I switch my Wii U into Wii mode can I still download games like Mario RPG? And can I use my GameCube controller and adapter?

You can download games that were released on the Wii shop.They will play as they originally played on the Wii.I don't remember if the Gamecube controller was supported for virtual console games.The Gamecube adapter works only with Super Smash Bros for Wii U.

But I just need to use the remote control for most of these games?

If Wii Mode in Wii U is functionally equivalent to an actual Wii (without GameCube controllers), which is what I heard, then you would need a Classic Controller for SNES and N64 games. I think all the other consoles supported the Remote alone (if I remember right, Genesis VC games supported the Remote but it was still recommended to use the CC to get a proper button layout.)
(and depending on the game, you might be better off using a CC than a GameCube controller anyways)