Provo has tennis lessons for kids and adults. For more information go here.

There is also a tennis league (Central Utah Tennis League) for ages 8-14 and 15-17. For more information go here.

Parks:

http://parkfinder.provo.org/ says that these parks have tennis courts

Kiwanis Park River View Park Rotary Park Sherwood Hillside Park Timp-Kiwanis Bounous Park

You can also play at Provo High, and at the elementary school on 200 N 600 E, and at BYU. At BYU they are pretty strict about the kinds of shoes you wear so as to not damage their courts.