Goblin Valley State Park is known for its Goblin Valley... most people drive in, walk around the Goblins and leave. First, you can plan for a longer hike since there are 3 valleys of Goblins, but I highly suggest you explore the other side of the park which is also totally spectacular and much less crowded. A beautiful bike trail system was developed a year and a half ago and seemed to be barely ridden or hiked. We were there on Good Friday and by 9:30, the Goblin parking lot was full, but the parking lot for this little gem of a trail system remained empty all day (we only saw two persons on the whole 12 km ride!).
If you don't want to bike the whole thing or if you'd rather hike it, I highly suggest you make a bee line to The Dark Side of the Moon trail that leads you along the San Rafael Swell. It'll be a 3-4 miles hike well worth it and you'll likely have the place to yourself (a good plan on busy weekends).
I suggest you camp just outside Goblin Valley in a beautiful and free BLM called South Temple Wash. You could come see the Goblins at sunset when it's less crowded (the best light, since the morning sun is lighting them from the back, as in my pictures, not ideal).
I've found out about these trails on the Trailfork app (I highly recommend you use it since the flower petal like trail system is a bit confusing even if there are signs at most intersections, there are lots of Do not enter sign that could keep you going in circles if you're not careful - we ended up doing some of the trails counterclockwise). I found this page to be the most informative about the trail system, even if it is a bit outdated, the info is still pretty accurate. Lots of sandy spots still (be careful of the sandy g-outs - sharp dips into washes - lots of them...). If you're planing to ride there, I suggest you start with the Buffalo Head trail clockwise - you'll be stopping all the time in the beginning to check out all the agates, jasper, geodes and quartz along the trail. If you're keep (and riding with kids), bring a small hammer to crack open some rocks and find geodes!
Then, skip the Landslide trail (the landscape is beautiful, but the trail is still pretty sandy). See this from the page mentioned above: While much of the trail is already firmly packed, there are long soft stretches which will tax even strong riders. The worst was a 0.4 mile stretch on the northwest side of the Landslide loop riding clockwise. Although it climbed only 100 vertical feet, the soft red dirt felt like grunting up Puke Hill. With the brakes rubbing. Into the wind. Towing a fat kid in a trailer.
Yeah, we got confused and rode it and it's pretty bang on.
Then, head to The Dark side of the Moon (ride it clockwise, even if there is a Do not Enter sign), the most beautiful trail that leads you along the San Rafael Swell (so close at times, that I walked my bike...). Then proceed to Desert View for more lookouts and if you feel comfortable with off-camber AND exposed trails, come down through Lizard foot (or do like me, and walk many sections!).