Rock Rock 4x4 Club hosts Easter Jeep Safari, which runs the week prior to Easter Sunday. Easter is April 8th this year.
There are trails in the Moab area that range from easily doable in a stock Jeep to pretty wild where large tires and lockers are requirements and damage is a real possibility.
If you go to the website you can read a summery brief on each trail.
I drove my JKU out last year, and the need to drive 2100 miles back was never far from my mind. With driving home whole in mind, we stuck to nothing tougher than trails rated 6 by Red Rock.
My Jeep has a 4.5" AEV suspension and 37's, with Rubicon oem D44's and the Rubicon lockers front and rear. Trails rated 6 were fun, but not a serious challenge, and posed no real risk of damage, absent a driver FU.
The event is fun, brings in a fun crowd and you can run on sponsored trail runs or on any of the many trails not utilized on a given day as you please. We did some of both, and had a blast. Beware that some sponsored runs can be a lot of hurry up and wait, with lines and back ups at obstacles. But that can give you an opportunity to get out and watch others takle the obstacles. Take the right attitude and the waiting isn't too bad.
I was back in Moab for a couple of days in August and we ran some trails then too. Lots of fun on your own as well.
If the Jeep and off roading are new to you, the EJS is a great place to run some trails that will be a lot of fun, but with support and suggestions should you need them.
BTW, my JKU is white with black too. Heres a photo from EJS to wet your appetite: