Being a musician on a budget myself I'll give you the least expensive way you can get decent recordings and if you want to spend more and upgrade you can later on. First thing you need is mics. Unfortunately getting good mics for drums can get very costly, being that you should have several mics, and drums are the hardest instrument to record. You can either buy a low end mic drum pack, probably a 4 or 5 piece that clip onto your drums, or you can get some condenser mics. I would recommend getting the condenser mics, look for used ones, SM57's are pretty much the go to mics for drum recordings. If you can get 2 good condenser mics you can set them up as overheads in an XY pattern and that will be sufficient enough for now.
Next of course is how you will record these mics, you can either record to your computer or to a separate device. Here is the factor, do you want to be able to record each mic as a separate track, or do you care if they all get recorded onto one track? Most devices will record onto one track, which is fine, but gives you much less freedom if you are planning to do any sort of editing. Unless you are willing to spend over $400 you will most likely not be recording in multi track. There are several input devices that will let you record onto you computer, but the main thing for now would probably be the number of inputs. You always want to buy something with more inputs, in case you ever decide to add more mics. I would recommend the Alesis USB mixers, there is an 8 input one for relatively cheap, this will allow you to go straight to your computer through the USB for recording. You could also get a standalone recorder, usually flash now, they let you record directly onto the device and later transfer them to a computer, I'm not too familiar with these, but you may be able to find a cheap one that lets you record in multitrack.
On your computer you need a program to record and edit your stuff, a free one is Audacity, for being free it is a pretty good program. Otherwise there are several you can buy;ie pro-tools.. So that's pretty much everything, I think? Some advice: good mics make good recordings, if you are going to splurge on one thing make it your mics. Also, good recordings are all in the prep work, make sure the sound you record is something you are happy with, never think you can edit it later to make it sound good. Also, always look for used gear first, there is no reason to buy new gear nowadays, look on craigslist or ebay, you can find some great deals and get very good equipment for a fraction of the cost. Sorry for the super long explanation, I could write more if you want!