WARNING: Major brain dump ahead. This should tell you pretty much everything you need to know about the basics of vinyl. (Note: I'm not the most objective person in the world (and a lot of this is just other audiophiles' opinions) and I'm a bit tired right now so take this with a little bit of salt.) If you just want reccomendations, scroll down a bit.
In vinyl there are 4 major components that are necessary: a turntable, a tonearm, a cartridge, and a phono stage. Usually the first 3 are bundled for tables sold new in your price range.
The turntable is what actually spins the records. It is comprised of the motor, the platter (what the records are placed on), and the plinth (the base that holds the motor, platter, and the tonearm). Its main contribution to the sound of a vinyl system is in the vibration dampening (both from the motor and platter bearings and from footsteps/speakers/etc.) and speed stability.
Turntable drive mechanisms come in 3 main types: belt, idler, and direct drive. Belt drives are the most popular among audiophiles. To avoid transmitting noise from the motor to the platter, they use a (usually rubber) belt to drive the platter. This is an extremely simple design to impliment, and so it's the most used today. This design has its disadvantages though: it usually has lower torque than other drive methads, belts wear out and require regular replacement (about once every few years), and less expensive units usually don't have very strong or well regulated motors (which leads to speed stability problems). Idler drive types are the next-most popular design amoung audiophiles. This type uses a plastic or metal wheel with a rubber coating on the top to spin the edge of the platter. These have high torque and generally have strong motors but also transmit more noise from the motor (because the motor is in almost direct contact with the platter). To my knowledge no company is currently making idler-drive turntables so if you want to go with one you'll need to buy used/vintage.
The least popular type amoung audiophiles (but most popular among DJs and audio pros) is Direct Drive. DD turnables have the platter either directly attached to the motor, or the platter itself is part of the motor. These are the hardest designs to get right but they can be very good when well implimented. Audiophiles don't like them (in general) because of their high background noise compared to belt drives and because they can exhibit "hunting" where the speed control mechanism constantly varies the speed by fractions of an RPM many times per second in search of the theoretical perfect speed. The more advanced DD turntables employ a Quartz-Lock technology that references the speed against a quartz crystal to get the perfect speed without any adjustments from the user (these turntables also usually use circuitry that reduces the hunting to the point where it is almost non-existant). Good Direct Drive turntables, such as the Technics SL-1xxx line and higher-end early-mid 80's Denons do a good job of dealing with the problems encountered with Direct Drive designs. Good DD designs feature high-torque motors with excellent speed accuracy and good vibration rejection.
The tonearm contributes mainly to toneality of the sound and the tracking abiltiy of cartridges. Unfortunately tonearms cannot be generalized very well (and I don't know much about them anyways) due to their reliance on everything else in the system and how they synergise with it. Generally you don't need to worry about tonearms until you get into the $700+ range, as almost all audiophile turnables in the $400-$500 range all come with a variation of the Rega RB250 which is considered the benchmark in budget tonearms, though it has some pretty big flaws itself requiring aftermarket mods to fix. One other major tonearm design you should be aware of though is the S-shaped tonearm (usually found on Japanese and DJ turnables). Many audiophiles don't like this design, prefering instead straight tonearms (NOTE: these are different from straight DJ arms (and NEVER, EVER use a straight DJ arm for listening to any record you consider valuable (straight DJ arms allow the tracking angle to vary wildly from the beginning of the record to the end, which means the record will have lots of wear except in one spot while straight audiophile arms adjust the angle of the headshell so the cartridge tracks at about the same angle throughout the entire record))).
The cartridge probably contributes the most to the overall sound of the system. There are two major types of cartridge: Moving Magnet and Moving Coil. MMs are usually high-output and almost all of them feature user-replaceable styli. They are also the least expensive type. They work by moving a magnet accross/between fixed coils. MCs are better sounding but also more expensive type. They don't have user-replacable styli because the coils move in relation to a fixed magnet (and of course the coils are attached to the cantilever and also wired to the output pins that are fixed in place on the back of the cartridge). They also usually have low output compared to MMs, though this isn't always true (the Denon DL110 has a 2.2 mV output).
Styli themselves are very important. Conical or spherical styli are round in shape, don't sound the best, track poorly, and wear down quickly (you should replace them about every 100-300 hours) but also do not require precise alignment. They're mainly used on DJ/Disco cartridges because they don't cause as much damage as other stylus profiles when they're out of alignment or used for back-cueing and other DJ'ing tricks. Eliptical styli are cut more precicesly and track and sound better than conical styli. They require more precise alignment though, and can cause more damage than conicals when they're out of alignment. When in alignment they can last for about 300-500 hours. Fine line styli track and sound even better but again require more precise alignment. They last about 500-800 hours when properly aligned. There are many more shapes that are even finer than fine line which track and sound better, but require even more precise alignment than fine line and can cause lots of damage when not aligned properly. When they're aligned properly you can expected them to last 800-2000+ hours (depending on the exact type). Conical are the cheapest, eliptical are a bit more expensive (and well worth the cost for their improvement over conicals in hi-fi setups), fine-line are more expensive yet, and other radical shapes the most expensive (they're usually reserved for mid-range ($500+) and above cartridges, though there are a few exceptions (such as the Ortofon OM-40, which has the FG-70 stylus profile at only ~$250)).
Matching cartridges to tonearms is very important for good sound and tracking. The main thing to pay attention to in matching (in addition to how they'd sound together) is the resonance frequency which has a big impact on tracking ability and how good it'll sound (though it won't tell you everything). If you're considering buying a cartridge it's a good idea to look up that cartridge at
www.cartridgedb.com and use the resonance calculator there (click on the "dynamic resonance" number for your cartridge to bring it up) which will tell you the theoretical resonance frequency. If it's between (IIRC) 8 and 13 Hz then your cartridge should (in theory) be a good match with the tonearm (as far as tracking goes at least). That isn't to say that it must be between 8 and 13 Hz to work, but just that that is the optimal theoretical range.
The phonostage has the most influence on how refined a vinyl system can sound and also influences the overall tonality, soundstage, and dynamics. All different designs of preamps are available at all different prices. Solid state preamps, though, seem to give much more return on the dollar especially at the lower ends (I can't think of any tube phonostage you can find for less than $200, though again I haven't really researched them in depth). An excellent alternative to seperate phono stages are those built into mid to late 70's and early 80's receivers. Some of them are quite good, and can be had for as little at $25 if you're patient and shop around.
One of the most important aspects of vinyl is how everything is setup. Even if you have great components, they'll sound like crud if you don't take the time to set it up properly. For an excellent article on how to do this see
this article. Some tools that you'll need to ensure proper setup include a VTF guage (I reccomend the Shure SFG-2, which is about $20), an alignment protractor (I reccomend the
www.turntablebasics.com mirrored protractor for about $20, though there are a few free ones online too (I reccomend
this one because it uses true Baerwald alignment, which the most popular free protractor doesn't)), and a level of some sort. A test disk is nice, but not necessary (and many people believe that they're only good for finding the resonance point for your cartridge/tonearm combo).
Reccomendations:
In your pricerange your options for buying new are a bit limited. The best you'll probably be able to do is to get a good deal on a Goldring GR-1 and a budget phono pre like the Musical Hall mmf Phono Pack for about $500. Another option (one that I would personally go for) would be to get a Technics SL-1200 MKII (
www.musiciansfriend.com has them for $400), an Ortofon OM-10 for about $60 (and this can be upgraded to an OM-20/30/40 by upgrading the stylus), and a DIY preamp or vintage receiver for about $50. You won't get the all-in-one package, but it'll probably sound better than the GR-1 + Phono Pack combo. The Technics SL-1200 MKII is built like a tank, extremely flexible (it even has on-the-fly VTA which most TTs at any price don't feature!) and has a good resale value (epecially if you only use them for home listening) so if you don't like it you can turn around and resell it pretty easily. If you feel comfortable with used I'd reccomend checking out a used Thorens, Dual, or Rega P2 (like the GR-1, but built with a better tonearm (the GR-1 arm is
nearly completely different from the RB-250), and I think (though I can't find my source right now) also has a better platter and platter bearing). If you buy used be sure the seller knows how to pack a turntable. Poor packing kills good turntables. Check out
this page for detailed instructions on how to pack.
Finally, if you'd like to learn more about vinyl I suggest checking out
this excellent vinyl forum.