Well, to hook up to a wireless connection, you'll need a wireless router (or something similar) to connect to. Most wireless routers come with ports in the back for wired computers too, so you won't have to make all the computers in your house wireless.
Then you'll need a wireless...attachment....to your computer. You can do the card option, which installs like any other sound/graphics card. It's probably the best choice to go with a card, but you can also use a USB (usually USB 2.0) hub kind of thing. You might want to check out the specs needed on whatever card/USB thing you get. The computer-side piece of equipment will (usually) come with a CD with a setup on it, so no worries there.
It's probably best to get matching brands for your computer side and router wireless parts, as they tend to go together better. Plus, the software will typically be able to automatically recognize your router if it's the same brand.
I'm personally using the Linksys company equipment. Unfortunately, I found out that the chipset they use isn't currently supported by Linux, because they're so protective about their hardware secrets, so I'll have to wait a while if I want to switch.
(For the record - A 5 year old computer is fine. I'm running Win98 on a piece of....junk, and it runs perfectly. XP is even better.)