A tripod is really two pieces -- the tripod itself, and the head that the camera connects to and lets you aim the camera. They need to be considered separately.
The tripod's purpose is to be a rigid platform -- but portable. you end up having to make a series of compromises around cost, weight and rigidity (choose two....). Cheapest is steel, generally rigid but heavy. Then aluminum, lighter but more expensive. Then carbon fiber, much lighter, but you either pay a lot more or give up rigidity (or both).
The worst tripod is one that's so heavy you won't take it with you. The second worst tripod is one that's so twitchy it doesn't really serve its purpose as a rigid platform. That said, you can pick up some decent tripods at good cost.
I started with an aluminum tripod, upgraded to a moderately priced carbon fiber. The weight difference was about a pound, which doesn't sound like much until you haul it around for eight hours. Then the cost of a carbon fiber seems like a bargain.
Wood tripods are fairly rare now; also fairly expensive, but very rigid and heavy. For large camera rigs (medium format, for instance) useful, for 35MM, not necessary.
You want a tripod that puts the camera at a comfortable shooting height. If you aren't comfortable using it, you won't. Less expensive tripods have shorter legs and center post that raises the the camera to eye level. More expensive tripods will meet closer to eye level. That center post can cause vibration and cost rigidity (especially in a windy environment), so if you can afford it, go for a taller one.
A center post with a hook on the bottom allows you to add weight to the tripod which will cut the vibration and add stability in wind. You can use your camera bag as a weight.
Most tripods come iwth legs that collapse in 3 sections. Some are in four, a few five. The primary advantage of the more sections is that the tripod will collapse into a smaller lump which makes it easier to store and travel with. More sections means more moving parts and generally a loss of some rigidity and added cost. I'd lean towards a four-part leg unless you're backpacking, but three is fine (especially if the tripod lives in your trunk).
Tripods have a weight rating. The heavier the capacity a tripod can carry, in general, the more rigid it is. You can put a heavier camera on a tripod than it's capacity, but you might end up fighting sag and vibration. I use the weight capacity as a rough estimate of rigidity when evaluating tripods, so the higher this value the better (but as you beef up the tripod, you're generally adding weight and cost).
Better tripods can give you flexibility on positioning -- if you do macro work, being able to adjust legs or shift the center post can be helpful (Gitzo tripods are good at this, for instance).
For someone just starting out, I'd recommend an aluminum tripod; it'll be a good starter unity that won't cost you a lot until you know what you really need, and not expensive enough that you feel too bad upgrading when you're ready. I'd plan on paying $100-140 for the legs to get into a price range with good quality and rigidity without paying for capabilities you find out you don't need.
Tripods like the Induro AT114 ($125) or the SLIK PRO 700DX ($100) would be good starting points. If you want to start with carbon fiber, the Slik 614 ($224) is a good starting point.
You then need a head on the tripod. It's purpose is to connect the camera to the tripod and allow you to aim the camera but hold it in place once it's where you want it to be.
There are basically two main styles of head: pan head and ball head. Pan heads use multiple hinges to allow you to adjust the camera in different planes. ball heads use a ball in a socket (similar to your shoulder) so that you have freedom to adjust the camera with a single locking mechanism. A third mounting mechanism is the gimbal, used primarily on really large (500mm and larger) lens.
I much prefer ball heads. You'll need to decide over time what you prefer, but most pro photographers rely on ball heads for most purposes.
Heads have a weight capacity just like the tripods do. If you put a heavier camera on one than it's designed for, you may find it sags or won't stay in position. You shouldn't overpay for a big head you don't need, because it'll cost you money and weight. But if you get one too small, you'll end up frustrated trying to make things work reliably.
A question with heads is "quick release or not?" -- there are a number of quick release options available where you attach a plate to your cameras and it allows you to attach and detach the camera swiftly. You want a quick release system, bceause screwing your camera onto tripods and taking it back off again gets old quickly. Which one depends on which head you want to buy. There are a few standardized setups for quick releases, the one most pros seem to use is the "arca-swiss", but it can also be pricey. I've standardized my quick release on the manfrotto RC2 plate, and it works pretty well; it's not as rigid as the arca-swiss mount but it gets the job done pretty well. I've only found a couple of situations (like my 12x100 astronomical binoculars) where the weight creates problems (those binoculars are huge and tend to be aimed at awkward angles). One reason I like the RC2 -- it has a switch that lets you lock the quick release in place. there's nothing quite like having your quick release release at the wrong time...
A good starting point for ballheads would be something like the Manfrotto 495 ($85). it's a solid basic unit. My primary head today is the Mnanfrotto 498 ($130), one step up from that. It's solid and reliable. I do plan on upgrading it (to a ballhead from Really Right Stuff with an arca-swiss mount) at some point, but that's a fairly major financial investment.
My current tripod is a slik 3 piece carbon fiber with the manfrotto 498. It's only major flaw is that I get a fair amount of vibration in windy situations because I have to raise the center column to get to eye level. Weighting the center column helps, but moving to a beefier tripod is the long term change at some point (on the other hand, switching to a higher end carbon fiber tripod and the RRS ballhead is much closer to $1000 than $100. It's an investment purchase).
my first tripod was a slik aluminum tripod with a manfrotto pan head. It was a good, solid starter tripod and I still carry it and use it for my spotting scope or if I need to have two cameras on tripods at the same time. It is (amusingly enough) more rigid than my 2nd tripod -- but a lot heavier. So it lives in the trunk. It would make a good starter unit for any photographer taking this step and the equivalent current products are listed above (the only items I listed I don't own are the induro tripod legs, but I've talked to enough photographers swearing by them and the really right stuff heads that I trust recommending them and they're the units I'm planning on purchasing next)
Don't overspend to start, because you don't need to. Don't underspend because you'll get something that you'll have to fight to do what you want. And remember that since the tripod and the head are separate, you can mix and match -- and you can upgrade each piece separately as you find out what you need and want.