There are a variety of options available. There are a number of beginner/intermediate loops cross country lower down on the mountain. The trails aren't very challenging, but they are very well constructed singletrack.
There is also a beginner terrain park with some log ride obstacles, jumps, berms, and a small wooden table top.
And then there's the Gondola summit trail. The first 2 miles are sweet singletrack with some sweet berms, the next mile is decent doubletrack, and then the last mile mile and a half or so is just a gravel road. Its a pretty steep climb, and especially on the road there is a lot of people, but the views are worth it! And its definitely a good ride back down... thats why you're sweating buckets on the way up, right?
Its Jackson Hole... Follow the sings from Jackson towards Teton Village
http://www.singletracks.com/item.php?c=1&i=2767