Use any oil you would like, they're literally all the same. Synthetic has it's advantages, but you aren't going to see any of them in my opinion unless you've swapped in a Ferrari engine (or some other high performance engine).
As for how often, I would say every 5k miles. The 3k mile thing is B.S. and has been debunked. I change mine every 5k and check it every 1000 miles or so just to make sure it's not low. You'll particularly want to check the oil regularly on your 3.8, as those engines in the earlier JKs are known to burn a lot of oil.
But in terms of oil, take your pick. Valvoline, Penzoil, Castrol, etc. Literally all the same and you won't notice a single thing from running any brand.