Not to be totally rude here. But I assume you bought the Wii new, and it came with a manual. I just don't see any point in typing stuff out that you can actually read in the manual. It explains all the questions you asked. It will also help trouble shoot your internet problem.
Are you sure you have "wireless" turned on on your router? Id check that first.
Sorry don't take my post as an insult. But the manual actually does a real good job explaining the many features of the Wii.