Nice job, Derby ... The power coming out of the transformer is probably 12VAC, depending on your system. It steps down the 110VAC to a lower voltage. You can use a volt meter to confirm this. If your 110VAC receptacle has no power ... well you might have gotten lucky! Solve that problem and you could be good!
Depending on the size of your lighting design, you might have one or more zones being fed by this transformer. A quick look inside will reveal the power connections. Again, your volt meter can show you how much voltage is being fed into each zone.
Start at the closest light fixture and check if it is receiving power. Frequently, the problem is only burned out bulbs. Plan on buying a fist full to get the system operating again. I recently replaced 75% of the bulbs on a whole yard system. Typical clip on power leads can be a pain in the arse ... but checking for sufficient voltage is your key. Some lights are hard wired with wire nuts or heat shrink tubing. Make sure these have not corroded or gotten water inside. Sloppy connections are a common reason for lights not working.
Work systematically thru your layout until you are sure the power and the bulbs are working.
Here is a helpful video to get you oriented