Hi, Nathan! Welcome to a world of awesome.
A couple of comments:
- Yes. Sheldon Brown knows everything. He died a couple of weeks ago, but he left us decades of distilled knowledge on his site.
- Figuring out your gearing is easy, even without the calculator Jive Turkey sent you to (which is useful; don't get me wrong). All you have to do is count the teeth on your chain wheel (it'll probably be 46 or so on the big one, which is the one you want to use) and something like 13 on the small one in the back. The ratio of those will give you your top gear. Then count the teeth on the big gear in the back. The ratio of the big front ring to the big back ring will tell you your low gear. If you want to climb hills more easily, you probably need a lower (that is, larger) large rear sprocket. If you want to have a higher top speed, you probably want a higher (that is, smaller) small rear sprocket.
- "Speed" can mean a lot of things. It could mean "acceleration", and the issue might be that you're shifting up too soon, which means you can't get pedaling very fast. Your only tachometer is your own nerve endings (unless you get a fairly fancy computer, and I wouldn't). It could also mean that you find yourself pedaling uncomfortably fast to get up to the speed you want. If the former is the case, you just need to pedal faster in lower gears, then shift when you redline your legs. "Speed" could also be a factor of efficiency. See "tires" below.
- I usually run one big chain ring in the front and an appropriate but limited selection of gears in the back. I find that seven speeds gets me around very nicely where I live. Also, leaving three gears in the front means that you need something to take up the chain slack when you go to a smaller chainring, which means having a tensioner, and the easiest tensioner to get ahold of is your rear derailleur...
- I'm glad you're thinking about tires. They make a huge difference. Unless you're hitting dirt and snow, you really don't need tread at all; it just slows you down. The whirring sound you hear when you ride is your lunch turning into sound instead of forward motion. A quick rummage around the Internet yielded these tires. The grooves are fundamentally for looks. Somewhere on Sheldon's site is an article about hydroplaning, and if I recall, a road bike would have to be going 130 MPH to hydroplane. So that's not an issue.
As for where to get stuff, well, that's always a challenge. If you live near a university, there's almost certainly a good bike shop nearby. The question is if they're going to sneer you out the door. For some reason, a lot of bike shop guys think that their job is to show disdain for your bike. So ask around, find someone friendly and knowledgeable, and keep going there. Ordering stuff online is OK, but they can't give advice or give you used parts for ten bucks because it's what you needed. Plus, you don't pay for shipping and you'll get the stuff just as fast.
If you take some pictures of your bike, particularly the right (drive) side, we might be able to point out the stuff you need to pay attention to.