There are multiple routes to take with the most direct being Smokey Mtn Rd. I grew up in Page and did a lot of exploring around there in everything from a dirt bike to a 1 ton truck (and forgotten some of it).
If there hasn't been a recent rain you shouldn't have any issue. Expect some rough spots and washboards. I have gone out there in the spring after heavy rain and flooding before the roads were repaired and 4wd was a must.
Normally you could get around in a 2wd and many places a typical sedan.
Smokey Mtn is a steep climb at the switchbacks but I don't remember it being bad after that. Be sure to go farther east to Alstrom Point at the beginning of your trip for some good 1000ft cliff lake overlooks.
Verify your spare tire. Take plenty of water. Bring a good tow rope/strap just in case. Might be worth your while to stop in at the station on the west side of the road at Big Water for current road conditions.
Have fun!