When buying a drum kit there are many roads you can take to get to where you want to go. I know budget determines which road you take.
Their are some things you can get away with for cheap money and other things you cannot. Lets look at the price of an entry level kit compared mid level kits. For our purpose here as an example we will be using a 5 piece standard kit for our pricing. Entry level kits can be had for $500 or less with hardware and cymbals. If you go mid level you will usually spend $500 to $700 just for the shell pack.
First lets look at the most important component in any drum kit that in my opinion there is no way around and the most important pieces of any kit. the cymbals. I'm going to refer to Zildjian cymbals here as they are my personal preference to my liking of what a cymbal should be and sound like there are other companies out there that also put out fine cymbals like Paistie, and Sabian, but I'll use Zildjian in our example here. Zildjian now has many lines you can buy in packs from starter entry level packs to pro level. With cymbals you get what you pay for. The cheap entry level packs like the Zildjian ZBT that can be had for around $250 you get 4 cymbals in the pack, 2 hi hats (bottom and top), a crash, and a ride. Problem is they sound like paint top cans and if I were to gig with any half decent musicians I would feel embarrassed to show up with these cymbals, they sound that bad. Then moving up the ladder you go into the ZXT series, same deal 4 piece pack go for around $350, now your moving into hub cap sounding territory for $100 more than the ZBT's they really are not that much better, both lines are made mostly of brass and are pressed out of a machine.
From here as far as the Zildjian series go you make a huge jump to the A series (these are my preference). These are pro level cymbals that have been around for ever formal known as Zildjian Alvidas. They are US made and use a mixture of alloys, along with hand machining and hand hammering to tune the cymbal, they get their unique sound but this extra labor and alloy used for these cymbals come at a cost, you go from $350 of the ZBT's to $650 to $700 depend on where you buy them. But once you get into the A line you can be proud and fell good you are playing on one of the best, also at this level of cymbal the will grow with you, you will never need to upgrade them, sure as you go you might want to ad different sizes. You can go look at the Zildjian line here Zildjian cymbal site
Now that I got the cymbal thing off my chest lets look at the differences in entry level VS mid level kits:
Entry level: You get what you pay for, cheap wood shells made out of low quality wood 2 ply's maybe some 3 not usually best suited for acoustics, usually poor workmanship, low quality drum heads, cheap hardware, this includes the tom mount on the bass, the lugs used for the tension rods to tune drum heads, the drum hoops, and the hardware pack they usually will include which consists of cymbal stands, hi hat stand, bass pedal, and some include the throne. These sets can be up graded to sound allot better by just putting better quality heads on and playing around with the tuning, but will still never come close to the sound of a quality mid level shell.
Mid level: When you mid level you are approaching pro level. You usually have a choice of birch or maple shells usually 4 to 6 ply shells, Heavy duty tom mounts, heavy duty bass drum spurs, heavy duty drum hoops, nice workmanship, decent drum heads, I can't say you get a great snare with this level kit, a top level snare can easily run $400 alone. At mid level most kits are offered with out hardware or with hardware, none of them usually offer cymbals but if they do they are usually low end cymbals. So basically if you go mid level new kit, you'll end up spending $500 to $700 for shell pack, a mid level hardware set new will run around $350, the mid level hardware packs don't usually come with a throne so for a decent mid level throne will run ya around $100 to $130 then add the cymbals the A pack new runs about $650. If you add this all up your looking at $1600 to $1850.
Bottom line: If you are new to drumming and want to give it a try I would recommend you go through your local craigs list, find a complete set with everything for $250 to $400, bring it home and start to play. If after a month or so you find you like and plan to stick with it, I would first upgrade the cymbals, go to a music store and try different cymbals find ones you like then shop for used on craigs list and ebay, Just start out with the basic cymbals a kit needs Ride, crash, and hi hats, once you have your cymbals, start looking for a mid level kit new or used, then upgrade bass pedal, hi hat stand, cymbal stands, and if your not totally happy with your throne upgrade that, then do ad on cymbals, all in that order.
To get an idea on prices of new sets and hardwae this is a good place start looking Musician's Friend
You can look at my signature to see what I am now using, I had to sell my kit awhile back and am starting all over again. Soon I'll put a post up with pictures of my set up. But I am still building and upgrading.
Note: You might want to read the topic posted here "Tips and info on buying used drums"