Tune the drums. Getting the drums to sound as good as they can acoustically, before you record them, is the best place to start, no matter what recording set-up you have.
Have the drummer play to a click track. Metronomes are very affordable and worth their weight in gold. A studio environment will expose and amplify any changes in tempo far more than the live stage would. Listening to the playback of a song where the tempo lags on the verse and suddenly lurches forward in the choruses will sound worse every time you hear it, but you can avoid these roller coaster tempos by having the drummer play to a click in his headphones.
Put a dynamic mic on the snare drum. The Shure SM57 is versatile, durable and very affordable, which is why practically every studio from major state-of-the-art facilities to bedroom project studios usually has several of them at their disposal. You probably have a couple of them in your mic cabinet, so attach one to a mic stand and have it aimed towards the rim of the snare drum. This will pick up the snare and hi-hat nicely. Experiment with the placement until you get a nice balance between the two.
For the kick drum, choose a dynamic mic with a good low-frequency response. If your budget won't allow for an AKG D-112 (in the $200 range) there are several less expensive options on the market that will work just as well. In a pinch, the aforementioned Shure SM57 will even do a capable job. Add a little muffling inside the kick drum to tame some of the resonance or ring, and mount the mic on a stand just outside the front of the kick drum with the mic pointing towards the beater. Listen to the sound and experiment with the exact placement.
Set up a pair of small diaphragm condenser mics to the left and right of the drummer, about 5 or 6 feet above the kit. These will ably capture the brightness of the cymbals and give a nice stereo image of the whole kit. Mixing these tracks with the kick and snare will give you the flexibility to get a nicely balanced drum sound. The price of condensers has come down in recent years and affordable models like the Audio Technica Pro37 have surged in popularity.
Routing each drum mic to its own channel on your recorder, adjust the recording levels of the mics, making sure that none of them are so high that they peak and cause the dreaded digital clipping. Add EQ and compression to taste, and then arm each channel to record. When the drummer is ready to play, press record and wait for the drummer to deliver that flawless take that will elevate your song to a new level.