Where to find the specifications
Listed on the furnace
Somewhere inside the service panel of the furnace, there should be a schematic for the furnace. This may list the electrical specifications of the blower motor. If not, the blower motor itself will have a label on it. It should list the voltage, amperage, horsepower, etc.
Using an Ammeter
An ammeter is used to measure the amount of current flowing through an electrical circuit. You can use a clamp on ammeter to measure the amount of current being used by the blower motor while it's running.
Simply set the meter to measure current (A in the pictured meter), and clamp it on the hot wire feeding the motor.
This method may require working on or near live wires, please use caution and/or consult an expert to take the readings
Calculate Usage
With the above information and Ohm's law, it's easy to determine how much power the motor will use over a given time period.
Ohm's law says that Power (P) = Voltage (V) x Current (I), so we can determine how much power the motor uses at a single instant in time.
ex.
P = 115V * 5.2A
P = 598 Watts
Next we have to determine how many kilowatt hours the motor will use, which we do by dividing by 1000.
P = 598 W / 1000
P = 0.598 kWh
So for every hour the motor runs, it will use 0.943 kilowatts of power.
Total power/day = 0.598kWh * 24h
Total power/day = 14.352kWh
Total power/month = 14.352Kwh * 30
Total power/month = 430.56kWh
Finally, if you multiply this number by the amount you pay per kilowatt hour. You'll see just about how much it costs to run the motor constantly.
Cost = 430.56kWh * $0.07
Cost = $30.14 per month
Notes:
Results using this process will be an estimate only.
Values used are not actual values, they are example values only.
Calculations may be incorrect, nobody checked my math.
Disconnect the power
Start by turning off the breaker, and pulling the serviceman disconnect, which will typically look something like this.
This will insure no electricity is flowing to the condenser unit while you're working.
Open the unit
Next you'll want to disassemble the unit, to allow access to the electrical parts. This will vary from unit to unit, so check the owners manual for the procedure for your unit. Once you have the unit opened up, make sure to discharge the capacitors.
These things store enough power to kill you, so you don't want them to discharge accidentally.
Resistance is not futile
Once the power is completely removed from the unit, it's safe to start poking around (electrically speaking, don't go busting the refrigerant lines). Start by tracing the wires from the condenser fan motor, back to where they connect in the electrical box. There should be 3 or 4 wires. In my unit, I had Black, White, Brown, and Brown with a White stripe (your model may vary). To determine if the motor is good, you'll measure the resistance across each coil. To do this, you'll have to disconnect the wires, so the motor is no longer part of the circuit (make note of where the wires connected).
Typically you'll have 3 wires, start, run, and common (we'll ignore my 4th wire in this answer). Set your multimeter to measure Ohms, and start measuring. You're going to measure the resistance between each combination of two wires to determine what each wire is, and if the motor is still good. Let's start with Black and White...
Black -> White = 15.9
Black -> Brown = 35.4
Brown -> White = 51.2
Knowing that...
Common -> Run = Lowest resistance
Common -> Start = Medium resistance
Start -> Run = Highest resistance
We can determine that...
Black = Common
White = Run
Brown = Start
If we also know that the two lower readings should always add up to the larger reading, we can safely say this motor is still good. If you measure 0 or infinity between any pair, that means you have a shorted or an open winding and the motor should be replaced.
Repeat the same procedure for the compressor motor.
Shorts on the ground
The other thing you'll want to check for, is shorts to ground. Set your multimeter up to test impedance. Put one probe on the equipment grounding conductor of the feeder, and the use the other to find a solid ground on the motor. You may have to scratch some of the paint off, especially on the compressor. Once you've found a solid ground, measure from each motor wire to your ground spot. If the meter beeps or give a low resistance reading, you have a short to ground. As with the resistance test above, the motor should be isolated from the circuit when doing this test (once a solid ground is located).
Best Answer
"Upgrading" the circuit involves running a new cable (of thicker gauge). Basically you would be replacing a 15 amp circuit with a 20 amp, for a benefit of 5 amps but with all of the work of just running a new circuit (either 15 or 20) which would give you a total of 30-35 amps instead of 20. Because you also have two circuits, if one blows, only loose whatever was on that circuit instead of everything.
Running a new circuit is the way to go. As a commenter said, you can "balance" the load. Ideally you would not have the AC on the same circuit as your computer as the motors can put off electrical noise.
The gauge of wire you use will depend on the amperage and distance. If your room is upstairs with the bedrooms, code will often require an arc-fault breaker to be used.