hazel
Here's some information. What I am about to tell you is from my own experience so i know it works.
You need a broadband connection and a broadband modem. You need to connect the Wii to the modem either with a wire or wireless through a router. The Wii has wireless built in. You then need to go to Wii's online shop - I forget what they call it- and purchase (just a couple of quid) a special version of the Opera browser. Then you can browse the internet and read you emails etc.
Technical details about making connections etc are at the site I mentioned in my previous post which you already know about anyway.
Please ask if you have any more questions.
Can't help with the xbox.