Yo Thenights alive! Welcome to the board! The good ol' boys are busting your balls because you have asked a small question with a big answer. I will try to give you a general overview.
Over the years, a whole lot of systems have been used. As the years have gone on, the trend has been to use more and more mics, which means more and more channels. The minimalists, which includes me, believe that 4-6 mics are usually enough. Some old timers actually used 1, 2, or 3, but that is *really* old time.
Generally, the minimum set involves 2 overhead mics, one on kick, and one on snare. As mics are added, you get dedicated mics for highhat, floor toms, rack toms, and in extreme cases, right down to the cowbell. In the most extreme heavy metal setups, they have a mic for every crash, ride, and china. For a beginner, I would recommend 4 mics to start, and you can add mics as mics, channels, mic stands, cables, and space (money) become available. It also depends on the kind of music you are recording. Generally the heavier the sound, the more mics people use. The rest of this will mostly concern itself with 4 mics. You can get more complicated as you go. At the end, I'll talk about specific mics in various price ranges.
The majority of your sound will come from the overheads, which are critical. You will then add kick and snare to bring them out in the mix, according to your taste. Usually the overheads will be identical or very similar mics, panned left and right, as a stereo pair. They will also go into 2 identical (if possible) preamp channels. If you only have a couple of good channels, and a bunch of cheap ones, use the good ones for the overheads. They can be arranged as a coincedent (X-Y) pair, as an ORTF pair, or as a spaced pair. Do a search on these stereo mic'ing techniques, and you will find lots of stuff.
Snare- This is usually a single dynamic mic near the rim, angled down toward the center of the bottom snares. Some people use a second mic facing up from the bottom, with the phase reversed, especially if the floor is soft (carpeted), and is sucking up the sound from the snares. Often, for this reason, a drum platform is used, to reflect sound from the toms, kick, and snare back up to the mic array on purpose. Snare will usually, but not always, be panned pretty much to center.
Kick- Usually a dedicated mic is used here, one that reproduces low frequencies well and can take high sound pressure levels (SPL's). It is usually a dynamic, and can be placed just off the center of the outside head, in or near a hole cut in the outside head on purpose, or inside the kick just off where the beater strikes the inside head, depending on the sound you are looking for. If two kicks are used, you really need 2 mics, but if a double pedal is used on one kick, you can usually find a good spot for a single mic equidistant from the two beaters. Kick is also usually panned to center.
Ambient mics (room mics) - Some times a condenser is added to the array, either at the end of a tunnel built out from the front of the kick drum (not really a room mic, it augments the kick with some natural delay) or several feet from the kit. This picks up natural room reflections and augments the overheads, picking up the whole kit from a different angle.
Ok, now for some recommendations on specific mics. What follows is just my opinion, and everybody has one. First, I would stay away from most drum mic kits, which are designed to give you lots of cool-looking mics that suck. Note that for every mic mentioned here, there are a hundred perfectly good ones I didn't mention. Everybody has favorites, and there are a hell of a lot of ways to skin this cat. I personally hate rim-mount mic clips, which transfer rimshots and rumble really well, and give drummers with poor control many chances to smack your mics. Get one or two serious heavy duty mic stands for your overheads, a lighter duty boom, heavily weighted at the base, for the snare, and a good heavy mini-boom for the kick. For the heavy duty stand(s), try-
http://www.musician.com/product/OnStage-Stands-SB96-Studio-Boom-Microphone-Stand?sku=452082
Overheads- Most folks favor a pair of condensers for this:
Wicked cheap- Do a search on Naiant on this board. There are people here who know lots more about these mics than I do, but they swear by them. Behringer (yes, Behringer) ECM-8000.
Cheap- MXL603/604, Oktava MC012, Studio Projects C-4, Rode NT-5
Mid Priced- AKG C451, Shure SM81
To die for: Check Josephson C42,
Neuman KM84 (discontinued)/KM184, Schoeps CM6, Earthworks (several models- what those Behringers wish they were)
Snare- The good news here is most people use cheap mics for this because they work-
Pretty damn cheap- Shure SM57 (don't try to go for a cheaper substitute-just do it. It's been a standard for years, and has many other uses, especially electric guitar cabs), My favorite-AKG D770. There are bizzillions of other basic stage vocal dynamics that are good for this, including Audix, Sennheiser e835, Electrovice (EV), and many others
More expensive alternative-Shure SM7b, which is also good for many other things, Sennheiser MD421 or MD441. These are all multi-purpose bad-ass dynamics with hundreds of uses. MD421 is my first choice for snare in most cases.
Kick- This is where I would spend a little money, because good ones are about $200, and are generally better than the cheapies. Yeah, I know the Nady's look good- on paper. I wouldn't do it.
cheap- CAD KBM412, AKG D110, Shure PG52
mid priced- Shure Beta52,
Sennheiser e902, Audix D6,
Electrovoice ND868, AKG D112. I like the D112 for mellow jazz, classic rock, blues, and it rocks on standup bass. I like the D6 for metal, alternative, wherever you want some "crack" in your kick.
To die for- Elecrovoice RE20- Great on kick, but also on tons of other stuff. Another badass top-of-the-line dynamic. Kick is only one of it's many uses.
Honorable mention: AKG C2000B, C414 (any of them). The first one is cheap, the second a lot more pricey. These are multi-purpose condensers that I have found very useful on toms, congas, djembe, as room mics, etc. If you aren't picking something up, sticking one of those wherever you are not picking it up will help. Oddly enough, the C2000B is AKG's top recommended mic for toms, even over the much more expensive C414, and I have found myself in agreement with the AKG techs. Who would've thunk?
Addendum- a note on ribbon mics. There are 2 basic types- cheap Chinese mics, rapidly becoming popular with home rec'ers, and wicked pricey- AEA, RCA, and Royer come to mind. My experience with these on drums is nil, so I don't have much to say, except that lots of people on this board believe the newer less expensive versions have many practical uses on drums. Hopefully somebody who has used them for drums can tell you more. If anybody wants to send me a pair of Royers, I assure you I'll check them out- (in my dreams). Hope this helps-Richie