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Seems almost every time I go to start the car I get a click (from the engine) and no crank. Trying again a couple seconds later the engine starts fine.

Could this be the starter motor, starter solenoid, or something else? The battery is charged and good, all battery connections are solid.

 
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could be a bad contract somewhere. u could even check the spark plugs and clean the rotor.

 
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Distributor is less than 6 months old, spark plugs and spark plug wires were replaced last month.

I don't think it has to do with the ignition system, as nothing happens but a single click when the key is turned. It as if the starter is getting stuck while trying.

 
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well I had I similar problem, I have a 92 camry l4 le and it just turned off on me all of sudden while driving, I replaced everything but the coil and distributor. I looked around and found that the wire inside the distributor cap were being cut and moved by the rotor and making contract with the metal inside the cap. I guess u can check the wires outside and the ones inside the cap. let me know if that helps.

 
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This is a problem either with your battery connections or your starter. First check to see if your battery cables are tight if they are then it is time to get a new starter. The bad thing with the dreaded click is that most of the auto-parts starter testers will show the starter as good. Don't be fooled. Just replace it either with a new one or if the car has a ton of miles on it go used and save a few bucks. Make sure if you do go used that the part has at least a 30 day warranty. This will protect you from a Shotty used part.

 
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All the battery connections are solid, and I'm pretty sure it definitely is a problem with the starter. However replacing a whole starter is costly, so I'm hoping someone would know which component of the starter would be more likely to be bad.

I am able to replace what is needed myself, I just need to know which part needs replacing.