I’ve only been to Utö once in the summer of 2000. Utö is a place to relax, enjoy nature, rent a bike, hike, or go swimming. There isn’t much to do but it is very much the Swedish culture to just enjoy being outdoors.
When I visited, I took the pendeltåg (commuter train) to Västerhaninge and from there took the bus to Årsta Brygga where you get the ferry to Utö. The ferry schedule is here:
The SL card covers the train and bus, but not the ferry. There is also a nice beach (Årsta Havsbad) about a 10 minute walk from Årsta Brygga if you decide to take a detour and just hang out at the beach.
I believe that there is also a ferry from Stockholm in the summer, but I think it is very long trip.