The air conditioning (AC) went out at my house last week, and I learned some important lessons from the experience. I hope these help you:
 

1. Schedule at least 2 (better 3) repairmen to come out and diagnose the problem as soon as possible (all in the same day preferably). Try to get trustworthy repairmen recommended by friends if at all possible.
2. If the AC is blowing hot air, turn it off and leave it off, otherwise when the repairmen come, the AC unit may be frozen up, and they will have to make a second visit 8 hours later to diagnose it (which may mean they charge you for two visits–around $160 potentially).
3. After getting 2 or 3 estimates, if they are really high ($1,000+ for example), ask the repairmen how much to replace the entire faulty part of the unit (inside part, outside part, etc.)

 
I called an “emergency” AC repairman out first, thinking he could diagnose and solve the problem quickly, but instead he both couldn’t come ’til the next day, then when he did the unit was frozen up due to #2 above, and then when he did diagnose it, he gave me a very high estimate that turned out to be ripping me off by $200.
 
I thought he was trying to take advantage of the emergency situation, but since I hadn’t called out any other repairmen, I had to schedule visits for them which caused another day of delay. It was worth it, however, as the trusted repairman I brought out did it for much less (although the cost was still high).