The Rim and White Mountains and there are thousands of camping sites in the Coconino and Apache-Sitgreaves NFs. There are also a number of improved sites and private campgrounds. Since you live on the East side of town those are probably the quickest to get to for you but there are as many around Prescott and Flagstaff. By-the-way since it seems you are new here
, it is "Press-kit" not "Press-cott".
There are also NFS maps that tell you which roads you can camp on and how far back from the edge of the road you can go. Some, like Coconino NF you can download from their Web-Site. Apache and Sitgreaves are in my opinion, a little behind Coconino and do not have downloadable maps and their camping distance rules are confusing but as long as you don't try to camp a half mile off of the road into the forest you will not be bothered. You will need to be self contained for most all of the unimproved forest sites. By downloading or buying the MVUM (Multi-Vehicle Use Maps) maps you will get an idea of where to camp. A little hint. I don't go camping on Memorial Day, 4th of July or Labor Day as well as other holidays. The traffic is terrible and sites are at a premium beginning about 5 days before the holidays. There was a 12 mile back up yesterday afternoon between Table Mesa Rd and Sunset Point north bound.
I suggest that you take a trip, without the trailer, up on the Rim and scout sites before you go. Picking one out on the map and then pulling the trailer there can get you in trouble. Some of the sites shown on the maps are down roads that are rough, steep, mud tracked and overgrown with low pine trees. If you are into improved sites There are a number of them listed on Good Sam's site and several state parks like Fool Hollow Lake or Woods Canyon Lake that are next to a lake and great camping BUT, those need to be reserved in advance as they can be full through the Summer. Fool Hollow is one of my favorites.