So I tried to start my car today but the engine wouldn’t start. I have a 1998 Mazda Protege with about 103,000 miles on it. I can hear it trying to start but nothing happens. I don’t think it’s the battery because the headlights and radio turned on but that’s all I can deduce. I don’t use my car that often so the last time I drove it was on Monday and I didn’t notice anything unusual. It got a tune-up before I bought it from my parents in July and everything was fine then. It has been raining pretty heavily all day so is it possible something got wet? I googled and possible causes might be moisture in the distributor cap or a faulty ignition coil. I’m hoping someone who knows more about cars than I do will tell me that this is very easy and inexpensive to replace. Also that maybe little elves will fix my car in the night and I won’t have to do anything except leave them some milk and cookies as payment.
Maybe the fuel pump is toast. With my previous car, when I turned the key to the 1 position I could hear a slight whirring noise. I suspected the fuel pump once when my car wouldn’t start, but my brother (mechanic) turned the key to 1 and identified that noise as the fuel pump. So he knew it was working.
Could be the starter, too.
Fuel pump isn’t too expensive (usually sits on gas line close to the tank, easy to get to.) Starter might be worse. Some cars have them in very awkward places.
I wish the elves on you! sprinkles pixie dust and mutters incantation for you
Will the engine crank? If not, do you hear a click when you turn they keyswitch? Do you hear a buzzing noise?
My guess at this point is that the battery may be weak. You mentioned that you don’t drive that often, so your battery isn’t being charged that often either. Depending on the make, a starter could draw a hundred times as much electrical current as a headlight. The battery may be able to provide enough power to light the headlights but not be able to even turn the engine over.
Yeah, it cranks but won’t start. I would have said that before but I couldn’t think of the proper terminology. Time to read Cars for Dummies, I guess.
When you say you hear it trying to start, what exactly do you mean? Are you hearing the engine crank ( the normal rar-rar-rar sound)? or does it sound different? Does it sound faster or slower than normal?
The radio draws almost no electricity and the headlights don’t draw any where near as much as the starter. It is quite possible that the battery is bad. Also are the battery terminals clean, or are they covered in corrosion? How old is the battery? Being a 1998 car I would assume that it has been replaced at least once. A battery much over 3 years old is pretty much on social security.
While coils do go bad, they don’t go bad very often, and usually they tend to fail hot, not cold. In other words when you are driving on a hot day, not starting cold. While a cold failure is not unknown, it isn’t where I would put my money. As far as moisture goes, again this is possible, but again not my top pick.
My first guess on a no start is always the battery, the battery cables, or a fuse. If I just said put a battery in it for every time I was asked about a no start, I would be right maybe 75% of the time. 
Most auto parts store and a lot of shops will test a battery for free. You can check the fuses your self, and as far as the battery cables, you can clean them your self, if you don’t mind getting a bit dirty.
Sorry about the milk and cookies, but most of the automotive elves I know either way beer or Scotch, and in both cases pictures of dead presidents. Pictures printed with green ink. 
ETA: IMPORTANT! If you decide to disconnect the battery for any reason always disconnect the negative terminal first, and then the positive terminal. When reconnecting positive first, and then negative last. Also before you do anything, observe which cable is positive and which in negative (the negative cable is attached to the engine block and or the body directly, not to anything else. Make damn sure you get the right cable back on the correct terminal. If you buy a battery, make sure the terminal is oriented the correct way around. If you see a blue cable that is negative, if you see a red cable that is positive. If they are both purple, go fish.
An engine not turning over can be a lot of different things, as you’ve probably figured out if you’ve tried reading online about it.
You can also go down to your nearest Auto Parts store and get yourself some kind of repair manual for the car - Chilton’s or Hayne’s are what you’ll usually find.
Often recommended with your situation is to start with the battery (try jump starting it and see if it a.) works and b.) holds the charge) and work your way through the ignition system. We recently had to go through this with an older car and it turned out to be the ignition module (not the coil). Not a difficult repair at all and in the meantime we did some needed tune-up/maintenance before hitting the right spot. At least that’s how my SO justified ignoring my idea that it was the ignition coil, cause the other stuff was less expensive to try first. If he’d just put out the $40 for the module to begin with…
I tried leaving out beer. It only made the elf cranky the next day.
A 1998 model that’s still on its original battery - I also say it’s the battery. I don’t see your location, but if you’ve been doing lots of cold starts, I’d say your battery has lived a long, useful life, and is now done. This has happened to me in the past, too - it just doesn’t start, put a new battery in, whirbang you’re off like a rocket. Batteries are like tires and clutches - basically consumables.