Awesome, thank you for the thorough answer! Do you think a battery that has been sitting dead for multiple years is worth trying to recharge? I've been told I can spend about $10 to recharge it at a mechanic down the road. Otherwise, my cheapest battery is at Walmart and costs about $50. Do you think I can trust a $50 battery or a $10 recharge to tell me whether or not my starter/alternator is bad?

You could try for free by swapping the battery from another working car. If a battery has been sitting discharged for 2 years it won't be in very good condition, even if you can charge it. Irreversible chemical reactions will have been going on inside it for 2 years.

Replace the battery, or as @alephzero suggests for a temporary test, borrow a good one from another vehicle. You will need to replace the original battery in any event.

If you can get to one of them, both AutoZone and O'Reilly will charge and test your battery for free. They can tell you if it's bad or not, without spending any money.

That battery is dead. It has ceased to be. Bereft of life, it rests in peace. It has kicked the bucket, shuffled off its mortal coil, run down the curtain, and joined the Choir Invisible! This...is an EX-battery!!! :-) Take it out, take it to your friendly local auto parts store (saves having to make two trips), buy a good battery (NAPA batteries are pretty decent, IMO - better than Walmart at any rate), put it in, and things should be better. Whether you also have a funky starter isn't something you can diagnose until you've got a good battery in there.

The only downside to replacing the battery, rather than borrowing one, is that it is money sunk in a car that may have other, possibly fatal, problems.

@BobJarvis-ReinstateMonica - NO! It's just pining for a Ford.

@PatriciaShanahan As Matt says though, a number of places will do a free battery check. If the battery turns out to be OK (and as other answers say, I really doubt it is OK) then the OP can decide whether to throw money at fixes or not.

@DonBranson - badum-tish! :-)

I've seen where a dead battery will drain a significant amount of juice from a jump, since it's either trying to recharge while the jump is happening or the battery is shorted, causing the current to flow away from the engine. Replacing the bad battery is definitely the first step, or at least disconnecting it so it doesn't impair the jump-start.

$10 to charge a battery! That is ridiculous. Do you not know anyone with a trickle charger?

@GlenYates, I thought it sounded weird too, just what people told me. And no, not anyone off the top of my head. I'm visiting from outside the country, and also staying about 30 min away from where the car is resting, so a $10 charge just sounded more convenient than sending out a mass message for a trickle charger.

Thank you all for the comments! I'm a total idiot when it comes to cars, so I really feel a lot more educated now. @BobJarvis-ReinstateMonica, I especially appreciate you putting it into persepctive. I will probably swap out the battery with another from one of my parents' cars (assuming one fits), or go by a mechanic and try to get a used one as someone else suggested. I'll then see if there are any other issues.

Good for you. Ultimately it's an 18-year-old Aztek, so you can hardly mess it up any worse than GM did in the beginning. :-)

@Christian Most 12V car batteries are "the same" for the purposes of temporary testing. As long as the cables reach the correct posts and tighten securely, most batteries will "work" although they might not "fit." (Now, if we're putting a Ford Fiesta battery into an F-350 sitting at -20F, then maybe not.)

@DonBranson Dang I wish I could put a bounty on your fjord comment.