The main thing to do in the park is to take walk or hike on one of the paved or un-paved trails.  A good clean web-site for learning more specific details about Zion National Park is found here:   Zion National Park   

One of the real enjoyable experiences in Zion Park is to go horseback riding.  The concession is across from the park lodge and riding is available for an hour or 3 hours.  Reasonable prices.  The 3 hour ride gave offers a terrific look at the canyon.  65% of the riders in Zion are new and inexperience horseback riders.  Trails are fine, especially for those who are afraid of heights, and they meander around the bottom of the canyon.  Mix riding with some hiking, perhaps the emerald pools, and you'll have a great day!    

OTHER ACTIVITIES:  There are some good horseback riding opportunities outside of the park and there are some outfitters in the area that provide guided experiences such as hiking, mountain biking, rock climbing, rappelling, jeep tours, ATV tours, and more.   Some of the long-time companies that provide these services a linked below:

Zion Ponderosa
Zion Rock Guides
Biking Zion

THE NARROWS:  A lot of people enjoy hiking in an area called "The Narrows" in Zion.  It is easy to get to during the main season by riding the shuttle to the top of the main canyon to a location called the "Temple of Sinewava".   From here you can follow a paved path (about 1 mile) to the where the main canyon begins to narrow significantly.  From this point - plan to get your feet wet, or rent some water shoes from one of the local guide companies.  Generally you'll be walking in water depth about 6 to 18 inches.   You're walking up the river but this is probably one of the most beautiful places to visit in Zion (if you like the water).   Check with the visitor information center about this and other experiences in Zion.  Make sure the weather is good.  You'll enjoy this most from June to September when the water is a little warmer, but you can get water shoes and other gear to do this any time of year - from a local outfitter.  

East Side of Park:  Don't miss the opportunity to view some bighorn sheep in Zion National Park.  They often hang-out around the road on the east side of the main tunnel.