If this is the first time you've ever been to an off road park I would strongly suggest making sure you go with one or more people experienced in wheeling or are familiar with the park.
It looks like you have a Rubicon so you currently have a rig that's more capable than the driver. Not a knock by any stretch since I'm in a similar boat as you. Don't get cocky and take it easy and learn what you rig can do before getting crazy. As my buddy told me back several years ago "You have a rig that is more capable than you, learn to wheel it stock without lockers and allow yourself to grow into it."
If you've never been to an off road park, I can't say that I'd recommend the cliff's for the first time out. Not that it's a bad park but it has a lot of tight technical trails that can and will damage your junk if you're not careful. (Ask me how I know this.
) Don't get me wrong, it is possible to make it through there unscathed as my wife has in her old rubi, but you have to be very careful. I would warn you about the Illinois snot they have over there at the cliff's, but with it being so dry lately there are probably very few really muddy area's that will swamp your rig down.
Truthfully I might suggest hitting the badlands for your first time out to kind of gnash your teeth the first time out. They have a much more diverse selection of terrain for everyone's tastes, not to mention skill level. They have rocks, mud, sand and creek trails not to mention the trails (in my experience) are a little wider and more forgiving. Just something to think about.
Also a question for you. Do you have a recovery strap? It should be a mandatory item for anyone who plans to wheel their junk. If you don't have one pick one up, they're not too expensive. Just make sure it is a recovery rated strap with loop ends, not a tow strap with metal hooks or a chain. Either of those last two things are the worst things to have with you because if that strap or chain breaks, it becomes a flying projectile that can (and has) kill people. I picked up the straps I use at Menard's for about $15-$20, but you can find them at Harbor Freight as well. Also if you're wheeling at any off road park in the tri-state region you'll need a flag. You can pick one up at any part, but my 2 cents if you don't know if you're going to be wheeling alot, pick up a one of those orange bike flags from Walmart, K-Mart or where ever for $5 or $8 and wire strap it to your jeep when before hitting the trails. If you find yourself hitting the trails frequently, you can always pick up a flag at the park of your choice.