Persona 5 Strikers sadly do not have autosave and unfortunately, you can’t just save anywhere either. Strikers isn’t a short game by any means, so you’ll want to save your progress one way or another.
In the “jail” world, players will be unable to save their progress unless they reach a checkpoint, which is marked by a “C” icon on the minimap and they emit a blue ray of light. Checkpoints can be encountered after completing a section in the jail and after completing the dungeon and they let you return to the real world.
However, in the real world, you can just about save your progress almost anywhere (except during conversations) when the save prompt on the bottom-right side of your screen shows up. It’s Triangle on PS4, X on Nintendo Switch, and Y on the Xbox controller scheme for PC (aka the “special attack” key).