I really want to get into kegging, but it seems to be really expensive. I don't want to have to buy a kegerator to keep it chilled at all times and I've heard that once a beer is tapped it has to be finished within a day or two. Typically, once I start drinking my beer, it lasts me 2-3 weeks at least. Does anyone have any suggestions?
A tapped keg will only last 24 hours or so,
if you're using a hand pump to fill the keg with air. If you're using CO2, which pretty much every homebrewer does, it's not an issue. If it takes you 2-3 weeks to float your keg, and you're replacing the beer with CO2, it will last you more than long enough.
Don't buy a kegerator, build one! If you can find a cheap fridge or freezer (maybe even a free one on Craigslist) you can convert it for cheap. Just drill a few holes, build a collar, and install faucets. Honestly, the tap faucets should be the most expensive part of your kegerator.
Note that if you use a freezer instead of a refrigerator, you'll need an external temperature controller to keep it from freezing your beer. They are reasonably cheap.