There is probably an old capacitor in the electronics which dried out over the time and now takes a long time to gain the necessary power load. If you are familiar with electronics you can check yourself for capacitors which looks bloated or otherwise funny and replace them.

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).