I have a Carrier Weathermaker 8000 furnace (58WAV090-JC) with a Lennox condenser (HS26-042-2P). When I turn the A/C on at the thermostat, the blower in the furnace turns on but the condenser unit outside does not. I have taken the following troubleshooting steps:
1. Check contactors inside condenser unit: I get 240 volts to the contactors, nothing out the other side. When I push the contactors in manually, the compressor and fan turn on. I checked the low voltage terminals on either side of the contactor switch, and got no voltage reading. Conclusion: Compressor and fan work, but the contactor is not getting the signal from the thermostat/furnace to turn on.
2. Check the wires coming out of the thermostat: Put my volt meter on the red and yellow wires. With the thermostat set to off, I get 25V. Turn the thermostat on, I get 0 Volts. Conclusion: Thermostat is working properly.
3. Check control circuit board inside furnace: The red wire coming from the thermostat connects to the "R" terminal, and the yellow wire connects to the "Y" terminal. Putting my volt meter on the R and Y terminals, I get 25V with the thermostat off and 0 Volts with the thermostat on. Conclusion: Connection from thermostat into control board is good.
The wire going out to the outside condenser unit connects to the "C" and "Y" terminals. Putting my voltmeter on these, I get the opposite as above--thermostat off = 0 Volts, thermostat on = 24 V.

Conclusion: something in the control board is bad. Fix: replace control board.

Before I go out and spend a couple hundred on a new board, am I doing this right? Does my troubleshooting sound correct?

No. You should not replace the control board. It has nothing to do with the AC. You need to check the outside unit. Some have a safety that you may need to press in and reset. Also, the coil could be bad on the contactor. Turn off power on both units and read across the coil and see what you get. If it checks out good with 500-3000 ohms then check from the wire at coil to either low voltage wire coming into unit. If one circuit is open you have a tripped safety. One wire coming in will plug directly to coil and the other one will go through the safeties first. Here are some diagrams: https://dl.dropboxusercontent.com/u/8010163/AC%20Wiring%20Diagrams.zip

If you have 24 volts in Air Handler going out to compressor, and you don't have 24 volts at outside contactor coil, The control wiring between the 2 is open, if they were shorted, the 24 volt transformer/fuse will blow.
Check low voltage control wiring.
Fuse is not blown, as blower and thermostat still works

Sometimes the control wires get cut by a weed-eater or chewed in two by an animal. You can take it loose at both ends and wire one end together and see if you get continuity with an ohm meter. If you do then the wire is good. This really should be done first. I should have mentioned this before.

Thanks, guys. With your help I found it. There is some sort of pressure switch on the evaporator cooling lines that has a reset button on the top of it. I didn't notice it before--it was hiding down in the bottom corner, while all the other wires and circuitry is up on top. Once I pushed the button, everything started working again. You guys are awesome!

If the low pressure switch tripped it is because you had an air restriction due to dirty coils, or the coils froze up due to running too long, which is hard to do in summer. Make sure to check the filter and you may need to clean the coils and the blower fan (you have to remove the fan to clean it).

You need to call an AC expert technicians to repair your AC system. Always keep in mind to repair your AC or Hvac repair Orlando (http://www.myorlandoac.com/) from NATE Certified technicians.

Being, this is NOT a SALES site. This site is free, and the link you keep providing is NOT free.