Are there tennis courts and if so what can you tell me about them?

There are some tennis courts on the island. I think the name of the club is the San Pedro Athletic Club.
I saw a poster for it in the lobby of the Mayan Princess. You could email them and ask. I think guests of the Mayan Princess have free use of the courts.

When my wife and I came to Ambergris Caye this past January my wife and I used the Isla Bonita Tennis Club right outside of town (you can see it from the landing strip). I don't remember the price, but it wasn't unreasonable (about the stateside price of a vacation-length health club membership--plus, they seemed willing to negotiate the price). It contained a couple of nice courts, a gym that was just a bit larger than what you'd find in a nicer hotel, and a nice pool. If all you want to do is play tennis, maybe you'll think it's a bit steep. My wife and I, however, like to exercise every day, and running on those bumpy streets and narrow sandy beaches got tiresome pretty quick!