Around 3 hours, Give or take, to the north end at Grand Rivers. Another 30 to Turkey bay.
Take I-65 south to the Western Kentucky Parkway, take that all the way to I-24. Go 10 miles or so on that and get off at the Grand Rivers exit.
You can also go south to Hopkinsville and then west and come in the middle of the park to get to turkey bay, but its about the same amount of time.
Of course the park is darn near 40 miles long (to the bottom in Tennessee) so it takes a while to get around in the park, but it is a beautiful park. And there are a lot of very low to no maintenance roads in the park you can legally drive on, if you want some mild wheeling with no fee. Definitely print out the full road map on their website, because the park map there is wayy to small, and not very accurate (the turkey bay map is fine however).
Last time I went we spent a few hours trying to find the camping spot we wanted, but we forgot the big nice map, so we were stuck with the one page of a brochure map hey have there... Needless to say we never found the spot we wanted, but we did find and awesome spot anyway (3 miles by water from the boat ramp though). And got to do some fun wheeling just getting there.
The next day we found the road we wanted originlly, the sign was back off this road 15-20 feet, and grown over... It was labled as a higher maintained road than the one we where on, but it was just a grass track through the trees.
We explored it anyway but a mile in got stopped by a downed tree with no obvious way around... Im bringing a chainsaw next time and clearing that road, because the camping spot would be great, across sugar bay from the boat ramp, so we could launch our boats, and then not camp in the camp ground.