The quest for the ultimate tone is a constant struggle for guitarists. I've yet to meet a guitarist who is wholly satisfied with every aspect of their sound. Coaxing a great tone out of your guitar and amplifier is certainly a frustrating experience for most of us - even with great equipment, that "perfect sound" always seems to be just out of our grasp. Let's examine several ways of changing your guitar sound without shelling out wads of coin for a new axe or amp.
Changing Guitar Picks
One of the cheapest and easiest ways to tinker with your guitar sound is to experiment with different sizes and gauges of picks. Using a very thin pick produces a sound drastically different than using a heavy one. Many jazz guitarists tend to favour using heavy picks (1.5 or 2 millimeters) because it tends to thicken and darken up their sound somewhat. Thinner gauged picks tend to give guitarist's a brighter sound, although it tends to produce tone with a shade less depth (I personally can't stand really thin picks, but some people swear by them, as they feel it gives them much more speed). When choosing picks, be sure they are well-made, and the edges of the pick aren't unintentionally rough, as this can interfere with your ability to play notes cleanly. I have found Jim Dunlop picks to be of excellent quality, but there are many excellent brands of guitar picks available.
Changing String Gauges
Another easy way to alter your sound is via experimenting with different string gauges. A guitar strung with extra light gauge strings will sound completely different than the same guitar strung with medium or heavy gauge strings. The String Anatomy 101 website offers explanations of different types of strings. For links to various string manufacturer's websites, as well as online guitar string retailers, visit the guitar string links page on this site. (It should be noted that changing string gauges on a guitar generally neccesitates an intonation adjustment. You can learn more about how to do this by reading the Intonation FAQ.)
Tweaking Your Setup
Making little adjustments to your guitar's setup, such as clamping down floating bridges, or adjusting pickup height, can also make a world of difference in the sound your guitar produces. If you own a Stratocaster, the Strat Tips website offers some great tips on how to go about making these adjustments, and what the "optimal" settings are. You'll find a ton of other ideas for tweaking your axe in the Guitar Repair Archive.
If none of the above procedures offer a tonal solution drastic enough for you, you can always consider replacing a pickup or two. The problem is, there are hundreds of electric guitar pickups on the market, and it's hard to guess what a pickup will sound like in your guitar until you've already bought it and put it in. The Harmony Central site offers an excellent resource; the Electric Guitar Pickup Database, a collection of reviews by guitarists, on specific pickups. The archive is huge, and chances are, you'll find at least several reviews of the pickup you're considering for purchase.
Emulating Your Heroes
Sometimes, the best way to go about finding a sound that is right for you is to emulate someone else's guitar sound, and then, over time, make adjustments to it. Use the archive of famous guitarist guitar set-ups on this site to experiment with other guitarist's sounds, and try to evaluate what you do and don't like about each.
This should get you off to a great start in re-examining the sound you're getting out of your guitar. Remember - much of your guitar tone comes not from the guitar itself, but from the fingers in your fretting hand, and in the way you strike the strings with your pick. Adjusting the amount of pressure you're exerting, the type of vibrato you're using, the part of your finger that comes into contact with the string, etc., can affect your guitar tone almost as much, if not more, than any of the above suggestions. Good luck!