Just south of Holbrook pick up 377. Follow it to 260, then to 87. Follow 87 until you are close to Phoenix then pick up N Bush Highway. Follow it to Usery Pass Rd. Follow that to Brown Rd, then east to Rt. 88. Go north to park entrance. Note Brown road name will change to Lost Dutchman Blvd.
If you don't mind two lane roads and like the shortest route turn off I-40 east of Grants and go south on 117 to 36, then left on that road to 60. Follow 60 to route 88 in Apache Junction, then north to park entrance. This would be my preferred route.