I wanted to know if you buy the attachment that connects your Nintendo DS to your Nintendo Wii and if your Wii can connect to the internet, can you connect the DS to the internet?

I don't think so but I maybe wrong.

As for the ds connecting to the wii I only know one game that does that and it's done wirelessly and that's poke'mon battle revolution in the wii and pearl/diamond in the ds.
As for the ds going online you need to have
1.a usb port internet adapter for the ds connected to your computer so the ds can go online wirelessly
2. have a broadband or dsl wireless router

1 & 2. you need the correct hardware for the ds/ds lite meaning either the ds internet browser or the ds lite internet browser. If I'm not mistaken you should be able to buy either one from nintendo.com but you have to have the corresponding ds system for the browser though.

Good luck!