I noticed that the air coming out of the vents smelled kind of moldy, like how an old car's air comes out. I can tell because the air in the house smells a bit moldy as well.

I got 5 houseplants because I read that they improve air quality, and that have quite a bit. But the moldy air still comes out.

I don't know how to fix this or who to get to fix it. I living in an apartment so I can't do it myself. What needs to be done to get the air to smell fresh and clean coming out of the vents?

I suspect that mold is collecting in the condensate pan below cooling coils, usually because the condensate pan was filled and/or clogged, not allowing the water that collected to drain properly.

This needs to be checked and cleaned to eliminate the collecting mold, and insure the condensation is allowed to drain.

Any other measure will only be a bandaid or otherwise mask the buildup of mold.

If this checks out OK, then there can be mold collecting in the duct work someplace for some other reason. A good duct cleaning is then in order.