Expert Trials are harder than normal missions and are unlocked for the first time during Free Time 3.1, with more available during most free times after that. The level recommendations for them are usually in the high 30s to low 40s, meaning it's highly ideal to wait until a second playthrough to try them. Doing an Expert Trial will take up 12 hours of your free time. You can get them from an NPC in the Main Gate area of Akademeia, who has "Trial" above their head.
Note that Expert Trials are restricted to only the free time in which they appear (with a few exceptions), then disappear until the next New Game+. However, completing them will add them to the list on the Missions option on the main menu, allowing you to do them over again if you wish.