Not sure what route you are taking or when you are going. But I highly recommend that you pick up (online) the FunTreks books for the areas you will be passing through. Very popular with Jeep folks and other off roaders. We use them all the time and in fact I usually keep the Colorado ones in the car. In those books they rate trails Easy, Moderate and Difficult. You will be able to do any "Easy" trail with your stock LR4 and with sensible, albeit novice, driving. It is a great way to get started and you will be amazed at some of the awesome drives you can take that are "Easy". A stock LR4 will not have a problem with Moderate trails either once you have driving experience and occasionally a good, experienced spotter. You can certainly try Moderate trails on your own but there will be greater risk of damage or getting stuck. (Speaking from experience.)
The other things to keep in mind is that when you go off road it is all about slow and steady. "Slow as you can, fast as you must." This means that you may often find yourself going 3-5 MPH. This will have a significant impact on your road trip progress. Even at 15 MPH, taking a 45 miles side trip is a 3 hour deal whereas on the interstate it is just over 30 minutes. It seems my wife and I are very poor at estimating how long we it will take to explore a road/trail/area. We nearly always underestimate by at least 50%.
As for your route, I would definitely encourage you to minimize your time on the interstates once you get to the Rocky Mountains. There are a number of incredible alternate paved highways. I70 is actually very nice if you are in a hurry and are going the northern route. But if you have more time and can take a more circuitous there are some great options, especially in Utah. If you do come in from the north on I-70 then you will want to either head south through Moab or on UT24 towards Hanksville then picking up UT12 which is an incredible scenic highway. Southern Utah is amazing. If you go the southerly route you of course have NW Arizona but you may want to skim the southern most parts of Colorado through Durango, Cortez then toward Lake Powell and either heading north on UT95 to, again, pick up UT12 (there is a very cool Easy back road called the Burr Trail Rd that you can use) or you can head more south through Monument Valley and on to the Grand Canyon. It all depends on how much time you have!