Ok, cool, that is what I will do, then I will let you guys know.
The two of you have been very... very helpful!!

Thank you

I know what you have been through and I hope this works ,let us know, please , Mike

Sounds like you have no voltage. Are you sure you have the capacitor hooked up correctly? You will need a voltage meter, without one there isn't anything we can do.

So I bought a voltage tester... Seems to me that everything going to and from the relay has 120v except for one black wire that is 24v and one black wire that has 0v. So I'm going to test my breaker in the basement. I tested both of my capacitors ohms, and they both gradually increase-which is good, right?

It sounds like you have one leg of power that is dead,( blown fuse or loose connection, bad breaker) you should be reading 220 volts across the two wires going to the a/c from the breaker.

I'm getting that--so 120v to each side of the relay ---120v coming off form each terminal of the capacitor--120v going to both the condenser and the fan, all of that is constant even when switched off. Then when I switch it to cool the relay makes a noise and then I have 24v coming off from one of the black wires. But there is a black wire on the other side of the relay that always reads 0v even off and switched on.

Start at the incoming two wires , with the two leads from the meter put one on one wire and one on the other wire ,you should be reading 220 volts here, if not the you need to go to the breaker , with breaker turned on , place the two leads of the meter on the two wires leaving the breaker, you should read 220 volts, if not then you have a bad breaker.
The capacitor in the a/c will only have one side of the incoming power and that is the 120v. That you are reading when measured to ground.
You should have 220 volts to the contactor and from the contactor(both sides).

I've got all that... so it must be that either my compressor is gone or the fan.
On a dual capacitor, would the fan work without the compressor, and vice versa?
They both aren't working, that's why I'm thinking it's something in the relay.

The only part that would keep the fan and compressor from running is a duel capacitor because it serves both, that is to say the contactor is closed and you have 220 volts entering the contactor and leaving the contactor when measuring the volts across the two wires entering and leaving the contactor. Is there anything in the contacts, like a bug or spider that would prevent a good contact of the center bar in the contactor?

I thought I looked all over for junk in there, but I'll look again.
I read another forum that said it might be a high-limit switch, but I can't find it on my unit.
This is really starting to bug me, I hate it when I can't fix things.

There is no high limit , that's in the furnace. In your checking for voltage did you varify 220 volts any where in your outdoor unit? You mention 120 volts but there has to be 220v.

Yes, I was getting 120v on each the black and white to equal the 220.
So, I know that I'm getting the right power from my main panel. I just can't figure out if it is in th relay or something else.

WOO-HOO!! I figured it out!! After I turned the air on @ the thermostat I went out to the unit and started tapping on the relay switch and got some sparks and then it turned on. Now when it cycles, it doesn't turn on--I have to manually set the relay switch.
So I should just be able to order a new relay switch.
Thanks for all of your help guys, it was great!!

Some times the slide mechanism gets rusty and will stick, if the control voltage is low ,21 volts or less it won't pull in.
If bugs get in it it will not work as well. You just may have to replace it to solve the problem.

Thank you very much!!