The wii mode is like a different section to the main wii u section. There's a wii icon you click to go into wii mode and it takes about 20 seconds to load and then you use it like a wii. You then can download titles to the wii section of the wii u. If the title is also in the wii u eshop it's best to just get it from there but if you get a title from the wii section and it later comes out on wii u eshop you can download it from there for like a buck fifty if you want to access it from the wii u section.
It's a hassle but yes, you can download titles off the wii shop onto the wii section of your wii u, or you can wait for it to be released directly on the wii u eshop. The wii section has lots of VC titles but the wii u section does not. GC games aren't on either. Also handheld games are only on the handheld 3ds eshop