This isnt a problem at all. I am on the second floor typing this and my wireless router is 2 stories down in the basement.
I am going to assume here that you connect to the internet via a cable modem.
====What to buy
So basically what you need to buy is a wireless router, and a wireless card for your computer. There are many out there, and they will all basically do the same thing so just get the cheapest. (Linksys, DLink, Belkin, ...).
* There are different technologies out there, Wireless b, g, and n being the biggest at the moment. Avoid 'n' it is still under developement. 'b' should be fine for your purposes. 'g' will potentially offer more speed (The problem here is that the internet speed coming into your house probably isnt fast enough to utilize the extra benifits of 'g').
- The router is simple enough to install, just follow the instructions, and post here if you have any problems.
- For you card for your computer, you can get an internal card or a usb card. An internal card is better, but you will have to open up your computer to install it.
The only security problem with doing this, is that others can steal your signal and possibly read your traffic. To avoid this you will need to encrypt your signal. All routers provide this, just simply follow your instructions.
Other performance considerations. The wireless internet runs at the same frequency as many other appliances (2.4ghz). So if you use a cordless phone for example, you may lose your signal if someone calls. (Not cell phones though, their different). Also if you get poor signal quality position the antennas on your computer and router to be parallel (the best you can). This will give a better signal.