There was another recent thread about sprinkler systems that has some helpful information and dandorn has offered good advise as well. I was in a similar situation when we bought our house, didn’t know a thing about the system and the controller was very old (analogue).
I would start by locating your sprinkler valves, there’ll be one for each zone. They might be in a box in the ground with a cover or on pipes sticking up a couple feet out of the ground. Take a look to see if the wires are still in tact there, those wires run to your controller. Then find the valve that provides water outside, ours is in the basement and gets turned off when the system is winterized. Make sure that’s on, you can then test to see if the sprinkler valves leak. Follow the steps on your controller to run each zone (some valves also allow you to manually open them by turning a handle). If the sprinklers operate, great, take a look to see that each head is operating, there’s no puddling, etc. If the zone doesn’t turn on, there could be an issue with the controller, the wires from the valves to controller, or the valves themselves.