Possibly the trickiest to get to work well, definitely the hardest to mix well...
Simple, but limited use way to mic a kit, is to use a capacitor stereo mic several feet in front, or a pair of mics a few feet in front, picking-up the 'ambient' sound. Very useful when multi-track recording a band with limited equipment, or if only a small amount of 'fill' is needed in the final mix. Drawbacks (like feedback) are obvious.
Next step is when limited by number of mics, simply hook-up the kick and snare. Very much to be avoided, but the bare minimum to get away with (try to get hi-hat with the snare mic if possible).
Dynamic Cardioid mic, placed inside drum itself, pointing at the point where the beater hits the front of the skin. If you have no stand, rest mic on the blanket, but try to avoid this. Punch comes in at 80Hz, but to get the 'hit' of the beater, and to just improve definition, boost somewhere between 4-6kHz (varies according to kit). A cheaky tip it to attach some plastic (like a credit card) to the drum skin at the point where the beater hits; for that modern (K I C K) sound.
Again, dynamic cardioid will do fine, a little above, to the side and towards centre of the TOP of the drum. To make it phat, boost 90-140Hz, to get it to bite, boost somewhere about 3-7kHz.
not always mic'd (often covered by overhead mic) but if needed, same as snare (take care not to get in way of performer's hands). Boost 80-120Hz to increase the punch and 'size' of the sound. Stick hits are far higher up (in high mid) so play in that region if that's what you want. If rumble is a problem, EQ or gate off the lowest frequencies (as Tom has little sub!).
Basically, it's really hard to close-mic the cymbals so cover the entire kit with either one (high) overhead or a stereo-pair in front (but high and close). Cymbals are the bastard children of hell, so keep at a low level, and kill bass lots and lots to avoid collecting the drums just below the cymbals.
Usually collected by the overhead used for cymbals, but if it must be mic'd, use a cap or back-electret mic because of the better hi-freq. responce.
When close-mic'ing remember the importance of gating and compression (esp. on kick) to separate the different parts of the kit.