You should have 5 wires running to the outside unit. 2 big heavy, insulated ones plus a ground from the breaker outside by the unit, and a couple of little insulated ones coming out from the furnace. With the thermostat calling for cooling, the big ones should have 240 V between them and the little ones 24 V. If both pairs of wires are hot, check the other wires connected to the rely they go to. If they aren't hot, the relay is bad. Fairly cheap and easy to pick up at an electrical or HVAC supply.
If either pair lacks power, follow it back and find where the problem is.
To do simple checks like this you do need some tools. A test light, a meter, or a voltage detector might be the best place to start with. I came across the niftiest gadget for trouble shooting, a voltage detector. They work through the insulation of wires. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit. You do not have to open up housings and expose electrical contacts. You are looking at where your hand is, not where the meter is. Most people are capable of doing repairs and will get it going and not get hurt if they use a little sense. The voltage detector makes it even easier.