When inspecting my wheel after noticing that it isn't true, I discovered that the culprit is a broken spoke. I ride often enough that this is something I need to learn how to repair myself. So, I assume I can just go to the bike shop and buy a new spoke. Then what? I have no clue how to do this.

Spokes come in many sizes. If you have all of the broken spoke take it with you to your bike shop for sizing. Or take a complete spoke from the same side of the same wheel as the broken spoke.
Be sure you have a good spoke nipple for installing the new spoke.

You can also buy them by the box, most come 20 to a box for not a lot of cash. 2x on the good spoke nipples.

To me this is one of those jobs where there's doing it and there's doing it right.

1. Why do you think the spoke broke? Without seeing your bike I'm going to guess that it's a rear wheel left side spoke. Since the advent of 8 and 9 speed rear hubs, the wheels are dished so much that non-drive side spokes often don't have adequate tension. The result is they flex too much at the elbows and eventually break.

2. The obvious first move is to get a spoke the same size and replace the broken one. If you tighten the new spoke until the wheel is straight you will have returned the wheel to the state that it was in previously. Unfortulately, what it was previously was a wheel that was about to break a spoke.

3. Find somebody who has a reliable rear wheel. Ask if you can "feel" their spokes. That'll give you an idea of the tension you are looking for. You'll probably have to tighten every single spoke on your wheel to get it to match the good wheel. If you tighten each spoke the exact same amount it won't affect your wheels' dish so the rim will be centered between the dropouts. If you're a good musician you can also use the pitch test by plucking every spoke. The lefts will have a lower pitch than the rights, but all of the spokes on each side should have about the same pitch. True the wheel by tightening and loosening opposing pairs of spokes an equal amount and you'll be good-to-go.

4. An alternative is to find somebody who is good at building and trueing wheels and who has the appropriate tools.