So I thought I needed a new battery and installed one. Now the engine cranks a but but won't start up. It really wants to but won't. And I have to recharge the battery. I don't think the problem is the mass air flow meter, because I think if that were the problem it would start but then cough and stall out. What's the troubleshooting sequence at this point?

 

engines need spark, fuel and compression to run. any one missing, and it won't.

 

The car is missing what I think is the A/C compressor belt. Does this belt needed for anything else, and could it have some bearing on my startup problem? I'm just trying to rule things out. Don't know much.

 

You should provide more background, like, what happened? Was the car running and then stop? Did you get a non runner, what? If the battery died, and you put a new one? A background is important!
A/C belt is irrelevant in this car as far as starting is concerned.

 

As Pierce stated, you need fuel and spark for your engine to run. Start by checking there. Easiest thing to check on the fuel side is whether your fuel pumps are running. When you go to start the car and you move the key to the II position you should hear the fuel pump run for 1-2 seconds and then shut off. The fuel pump is beneath the rear seat on the drivers side. Listen carefully. If you don't hear the pump run when performing this test then you have an issue with the pump. Could be as simple as the fuel pump relay.

To check for spark, pull one of the spark plugs and connect it to the spark plug wire and hold the plug against bare metal on the engine to ground the outside of the plug. Use an insulated set of pliers to hold the plug and have someone crank the engine. You should see a spark jump across the gap of the plug while the engine cranks.

As lev stated, information leading up to the no-start condition is important in the diagnosis. For example, if you were driving and it shut off suddenly and now doesn't start could lead me to believe the timing belt broke. If the car has been sitting for nine months, then there could be a multitude of problems.