The Arizona Trail (AZT) is an 800+ mile thru-hike that stretches from the border of Mexico to the border of Utah. Part of that trail comes through Roosevelt Lake and connects five other trails nearby. There are hiking options around the lake and near the marina for hikers of all levels.
You can enjoy Cemetery Trail which is the final resting place of the builder of the Roosevelt Lake Dam, as well as the Three Bar Route which has an amazing fenced in wildlife area that is studied by Arizona Game and Fish. The Vineyard Trail 101 has amazing views of Roosevelt Lake, the Dam, and Four Peaks. Nearby you will also find great Arizona hiking on the Thompson Trail 121 which is right off Highway 181, or you can hike near Roosevelt Lake by visiting Cottonwood Trail 120 which you grab from the Frazier Trailhead and takes you through some lower ridges and drainage into Cottonwood Canyon.
Feel free to also explore other trails near the lake (Tonto Basis Ranger section) and then come to the Roosevelt Lake Marina Bar for a nice drink and some new friends.