Ah, the tripod. Nobody, but nobody likes to carry one. Yet nothing, but nothing, will improve your photography as much. The tripod is the single most important camera accessory most photographers can have. The most obvious advantage is that it gives your camera a stable mount from which you can use any shutter speed, no matter how slow.
For slow shutter speeds be sure to use a remote release to trip your shutter; you are wasting the stability of the tripod if you push the button by hand.
Another, less obvious, but very important utility to the tripod is that it lets you study the scene through the viewfinder and contemplate what you really want to capture. You can scan the edges of the frame for distracting elements, you can ponder the position of you main subject of interest, you can get a feel for the balance and flow of the image. None of this can you do when the viewfinder images is jiggling around with a hand-held camera. And if you really, really like the image you are making, with a tripod you can click off a few frames in a row of the same scene, each identical to the others. This gives you some insurance against the photofinisher scratching your one-and-only version of that fantastic shot. (If you are shooting slides, these extra copies of the same image are called "in-camera duplicates." These in-camera dupes will be of higher quality than any after-the-fact duplicate slide can ever be.)
Of course there are some venues and styles of photography where a tripod won't help you a bit. Crowded places and sporting arenas probably won't let you use a tripod, and with good reason. Those three legs splaying out are a trip hazard. Photojournalism, street photography, candid portraiture, these are styles of photography where a tripod will just get in the way. With these kinds of photography, capturing the moment and the event are more important than technical quality of the image. But even these shooters will find their photography improving over time if they can practice at times with a tripod.
Carrying a tripod is a pain, but I've learned over the years that it is much less painful to carry a tripod now than to mourn over missed shots later. You can make the tripod easier to carry if you use a strap to sling it over your shoulder, or pad the legs so you can carry it more comfortably on your shoulder. Simple foam rubber cylinders that slip over the tripod legs are available from a variety of sources: you can pay about $30 for a set through a photo store or about $3 for a set of pipe insulation at K-Mart. If you attach this insulation to your tripod only put it over the uppermost leg segment -- if you put it on the lower leg segments they won't be able to telescope back into the upper segments. The pipe insulation comes with a slit along the length of the cylinder so it can easily slip over the tripod leg; once on, wrapping the ends with duct tape will keep it on forever.
A sad truth of tripods is that the heavier, the sturdier. A $20 lightweight plastic tripod is mostly useless. If you're shooting in a studio there is no upper limit to how heavy -- and stable -- your tripod can be. In the field, there are limits. Shooting at the roadside out of the trunk of your car permits a heavier tripod than backpacking in the Sierra Nevada, but you still shouldn't get a tripod heavier than you will actually use. One trick for carrying a relatively lightweight tripod but shooting on a heavier one is to hang something weighty on the tripod while you're shooting. Some people keep a nylon bag with their tripod and fill it with rocks when they're ready to shoot. I drape my photo vest or photo backpack on some protuberance on the tripod head -- you're carrying all that heavy lens glass with you anyway, you might as well put it to good use!
If you are going shopping for a new tripod, be aware that you can buy the legs separately from the tripod head. This lets you mix and match to get just what you want. Tripod heads come in two flavors: ball heads and pan-tilt heads. Ball heads let you move the camera freely in all directions until you get the composition you like; then you slide a lever (or some similar operation) to lock the camera in place. Pan-tilt heads, on the other hand, have separate controls for each of the three axes about which your camera can rotate. You can loosen the lock on one rotation axis at a time to carefully fine-tune your composition. Which type is best for you is entirely a matter of personal taste. I started out with pan-tilt heads, but when I found I was leaving all three axes loose as I framed my compositions, effectively turning my pan-tilt head into a ball head, I switched over to using ball heads.
As you budget your purchase, figure on $100 or $150 for a good solid tripod/head combination bought new. I certainly don't know all the tripods on the market and there might be excellent ones that are less expensive, but I'd be wary of spending less. A tripod too much cheaper than $100 risks being too flimsy for high-quality photography. (Of course, you can always pay more! There are some truly superb tripods out there for over $500.) My mainstay tripod uses Manfrotto (used to be Bogen) model 3221 legs and a Manfrotto model 3055 ball head. In round numbers the legs cost around $100 and the head about $50. This model of ball head is considered too lightweight by many nature photographers, but I like it just fine. The 3221 legs are a workhorse for nature photographers, a lot of us use them. For backpacking I go with the lighter Manfrotto 3001 legs. If you go shopping for a new tripod I suggest you look at the 3221 legs along with whatever other else you and your salesperson decide to consider -- not that you should buy what I use, but it is a good standard for comparison. It's a starting point if you don't know where else to start.