Setting up wireless internet: components?

Hi, my knowledge of networking is very limited so I'm asking a basic question.

I already have a computer connected to high-speed internet using a Speedstream modem. It is connected with Ethernet to the modem, which has only one Ethernet port and one input for the phone line. Also the modem has a USB but I am not using it.

What I want to do is to get a second computer in the room next to where the first computer is, and to have access to Internet to both computers. I guess one would use Ethernet and the other (the new one) would have to use wireless.

From what I've read I need a wireless router, and probably a wireless adapter for the new computer. Microsoft say some new computers have an integrated wireless adapter: my new PC will be a desktop with an NForce 570 motherboard: I'm buying all the parts separatly and building it myself. Does it have integrated wireless?

I noticed prices for wireless equipement are so much cheaper on eBay.ca (I live in Canada) so after doing a little bit of research I was leaning towards something like this for the adapter and this for the router.

There are many different versions of this router so I don't know what's good or not.

I'm a student on a very limited budget so I'm looking to spend as little money as possible. I don't need extensive features, I just need the signal to get to the other side of the wall and have stable internet on two computers.

Thanks.