If you are simply aiming to get a sense of what your vehicle can or cannot do, then there are a couple off-road parks (private) as well as Mounds up near Flint and Silver Lake (sand dunes) in Mears, on Lake Michigan. These are good, reasonably safe places to get your feet wet, so to speak. Silver Lake is actually fun. Mounds, I've been there many years back, wouldn't mind going back, if only to watch the idiots break their vehicles in the mud pits.
The private parks I avoid. These parks are where most of the "wheelers" hang out anymore. I won't hide my dislike for this facet of the "sport" - I think it is asinine in and of itself, but more importantly I see that its popularity paints our activities as being separate and apart from broader issues of land access. Give the overgrown kids with Jeeps a hundred muddy acres and they'll be happy, the logic goes, and then DNR/NFS can close down the two-tracks through the state and federal forest land with impunity and without objection. But don't worry, friends, we'll still let the loggers and gas/oil folks have access when and where they want. This wasn't the case 15 years ago, or at least not anywhere near to the extent it is now, but this is surely the trajectory of things nowadays. The results is that those of us who want to explore the woods responsibly (i.e. on trail, tread lightly, etc.), camp and hike and fish away from the crowds, we are left SOL. But enough of my editorial.
If you're looking for bona fide back country stuff - trails alone and away from civilization that you can reasonably manage with a well-equipped but largely stock 4x4 - there are a few pockets in the northern LP and a fair bit in the UP, but that's quite a haul unless you're going to make it a multi-day event. Unless you intentionally set out to get yourself stuck or broken, or set out in bad weather, good driving and street tires can get you through most of this terrain, and mud tires and a shovel will get you through the rest. Most of Michigan's natural terrain (as distinguished from its off-road parks) is very accessible and requires little in the way of vehicle modifications, even on the rocky bits of the Keweenaw.
Honestly, this stretch of muddy hill outside Marquette is one of the more difficult stretches of legitimate road/trail you are apt to find. As you can see, one needn't buy all the crap out of a Quadratec catalog, or have a sky-high lift, to get through this area. So if you can navigate this stuff the way your vehicle is set up, you're good to go!