If you do go, I would avoid attempting the Black Bear Pass trail with the F150, even with 4wd. It is likely still closed in spring, as are most of the high passes. But there are plenty of nice off-highway drives that don't go that high. but the pickup trucks have too long of a wheelbase for many of the high roads. Switchbacks are brutal, and a short wheelbase 4wd is necessary. You can rent a Jeep in town from several operators, many of which are "you drive it". Keep in mind some of them ban you from Black Bear and/or some of the Alpine Loop trails.
Most of the high passes are open by July 30 or so. I have heard that they may close them early this year. There was a significant snow up there last weekend. If it sticks, or if there is another storm close behind, they may just choose to close them down early.