I have a 98 Plymoth Breeze and about 3 days ago the heat stopped working. I had this problem maybe like 2 weeks ago, so I put antifreeze in it and the heat started working fine. Now it just blows out cold air. There is no antifreeze leaking. What could it be?

Your thermostat could be stuck in the open position, this would cause the water to just keep running through the engine and not give it time to heat up, or maybe your heatercore is plugged making it so the water wouldn't go through it to give you heat. Just two different ideas to throw out there.

Someone told me that there could be air stuck in the coolant and that I needed to take the cap off and let my engine run for 15-20 minutes. Could that be a possibility to?

I don't think so, if you do that don't open the cap after your car has run because your system is presurized and if the water is hot you could get burnt not to mention antifrieze all over the place.

If you have to refill the cooling system and it's not leaking, where is your coolant going?
If you only had a faulty radiaotr cap, coolant loss would show at the overflow tank running over or at the drain tube.
Have the system pressure tested and/or block tested. If not caught soon enough, an undiagnosed cooling system failure can lead to expensive repair bills.

So I can't open the cap and wait for it to bubble?

The guy told me to open the cap where I put my antifreeze in, wait for it to start bubbling (he said it would take like a minute or two) and let my car run for 15-20 min. You're telling me that is dangerous?

Running the engine with the cap off is not testing the system under the conditions in which it is to operate. Plus running the engine till hot coolant blows out of the radiator can be very dangerous.
There are special testing tools that can be used to determine if the engine has internal/gasket failure. Find a qualified shop to diagnose your car. Ask family, friends or neighbors for references.

How much does it cost to get a computer scan on a car to figure out what wrong?

Look, I don't have any money and I want to find out what's wrong myself. How do I do that so that I can seek further help from a professional? The mechanic told me there could be air running through it. Can that be a problem to?

I don't think air is the problem, but if you open the cap wher the radiator fluid is added (should have a plastic cap on it) not the steel radiador cap you can try your theory. Also did you mix the radiator fluid 50/50 1\2 water and 1\2 antifreeze before adding it I think that is the way they still do it, its been awhile since I've needed to add any radiator fluid. Good Luck!