I started singing more seriously recently, and I've gotten my range to about where I'm happy with it (F2 to about a G5 mixed voice). I don't have any problem carrying a tune or singing in key or anything like that, but it doesn't really matter much because my voice naturally seems to sound like shit tone-wise. Is there anything I can do to fix that, or am I pretty much stuck with it?
The more you practice the better you get. If you can get a good vocal instructor you can make improvements much faster. Work on proper technique and you can figure out why you don't like what you hear and what you can do to sound better. I personally don't like my voice but it's gotten a LOT better since I started taking singing seriously a few months ago. My problem is technique and the fact that I always sound congested, but that's because of my allergies.
You can learn to alter your tone a lot by manipulating different parts that affect your voice like your jaw, larynx, tongue, soft pallate etc. Professional singers can fine-tune their tone this way. There is loads of stuff to learn. Get some lessons.