I have some rental property, where the first tenant 'rewired' the furnace for "me". He burnt up the contact switch in the a/c unit. The next tenant had her uncle come out to troubleshoot it. He removed the contact switch from the a/c unit, then decided he didn't want to put it back in. So, now, basically I need to know how to completely rewire the system.
I've got the new contract switch and capacitor.
Coming from INSIDE the a/c unit, I've got these wires:
Brown
Brown w/white
Black
White
"common" (thick black)
"Start" (thick blue)
"run" (thick yellow)
(there's also a transformer inside the capacitor/contactor compartment)
Coming TO the a/c unit, I've got these wires:
Red (one side of the 24v transformer)
Purple (hooked to the red on the thermostat)
Red(white stripe) (yellow on thermostat)
Black (120v)
White (common)
Green (gnd?)

What a mess. Those are not the wire colors I am familiar with. Are the terminals on the furnace lettered to correspond to the wire colors? If they are lettered more like as in the paragraph below, I would start over again. You don't need to replace the wires. Just make sure for instance that the Y terminal of the furnace has a wire going to the Y terminal of the thermostat and another one going to one side of the coil of the A/C relay. I do not understand why some HVAC systems have more than one transformer. It only complicates things. You should be able to make one work as long as it is big enough to handle the blower relay and either the gas valve or A/C relay.

Usually there is a 24 volt AC transformer in the furnace with the secondary winding connected to a red wire running to the thermostat and a blue wire, common, to the gas valve, A/C relay, and fan relay. From the thermostat there will be white wire to the gas valve, yellow to the A/C, and green to the fan. The thermostat is wired to switch the power from the red to the white, yellow, and green as needed with the blue completing the circuit. Most thermostats and furnaces have the contacts labeled R, B or C, W, Y, and G for the corresponding wire colors. It may be wired to have the A/C control wires return to the furnace and its controls and then a second wire goes to the A/C unit. Internal wiring may replace the green wire if the thermostat does not give you the option of fan only or continuous fan. Digital or programmable thermostats may need the blue wire connected to them.

I need to go work on some of my stuff now. If you can't make this work, I will try to help more later.

Ok, doing some research, I can up with an ALMOST complete picture (attached) of how *I think* this system should be wired, please tell me if this is correct, which connections are missing, and where my limit switch, and blower control should be wired in...

Your diagram looks good. Usually the yellow wire to the thermostat and the one to the A/C relay are both connected to the Y terminal at the furnace. It is turn is connected to the control circuitry, limit switches, and fan relay. It is common for the blowers to use different speeds for air and heat. It is also common for the blower to delay starting and shutting down. Fortunately, I have never had to dig in farther to get the A/C to work. I would hook it up just like your diagram except making sure the Y at the thermostat is connected to both the Y at the furnace and the A/C coil. If that doesn't give you blower, at least after a few minutes delay, as well as compressor, You are going to have to dig in deeper. A schematic or an installation guide would be nice. Usually one is shipped with the units, and often found inside. The web page, or a local dealer may help also.

Just getting as far as your diagram was a good piece of work from the mess you started with. If it works, great. If not, good luck.