Car only starts when it feels like it.

Here's the poop:
I own a 93 del Sol. I know nothing of mechanics, so go easy on me. I'll do my best to describe the problem using my non-mechanical terms

I drove home from a meeting the other day (approx a 60 mile drive). When I arrived home and opened the car door, I could hear a hissing noise - almost like air escaping. My friend suggested maybe it was the a/c compressor (or something close to that). I turned my car off and then restarted it to see if I could hear it again. My car turned over...went like it was going to start and then died. Tried it again. Same thing. Third time, it started.

To make a long story short, here's what I have figured out. If I start my car cold, it starts with no problems. If I drive my car even just around the block, turn it off and then try to restart it, it takes several tries. I noticed that if I listen when I turn the key, I don't hear the fuel pump prime (again, I don't know if that's the right term...but I think I'm close) the times it doesn't start.

So.... what am I in store for here? Is this going to wipe me out financially? Is it most likely going to be the fuel pump or something along those lines, or could it be an indication of another type problem?

*sigh* It always seems to be something...

Mary