From Apache Junction you can take Hwy 60 through Superior/Globe and to Show Low. There are some steep grades but many RVers do it, as we have multiple times. It just depends how your motorhome does on uphills and downhills - or how you do
Alternatively, you could take the Usery Regional Park road (207 I think) from Apache Jct. which will take you to Hwy 87 and north into Payson. You'll have a steep uphill to Payson but as soon as you get there Hwy 260 eastward isn't bad to take you to Show Low.
Try to do either drive mid-week. There are a LOT of RVers going & coming both routes during the weekend.
Once you get to Show Low take 77 north and pick up 180 before I-40 and drive about 20 miles on 180 SE to the Petrified Forest Rd.
At the intersection of Hwy 180 and the park road are two gift shops. The one on the south side of the park road is the Crystal Rock Shop (I think that's the name). They welcome overnighters for dry camping and they even have a couple electric hookups. We've stayed there a few times.
Then you could drive the Petrified Forest park road in your towed vehicle or if you don't want to stop overnight you can easily take your motorhome on the park road for a drive-through tour which ends at I-40.
Have a good trip!