Hey.....I know you will hear about this from the doom and gloom crowd that will insist upon spending your money for you and surely not coming out to help you do the job. SO.....here goes.........
Am in almost the same exact situation. Goat barn/shed 300 feet from the house. Needed lights, fence charger, outlet for hair dryer for the kids, heat lamp, water warmer for low 20's, and maybe a charger for spotlight..............
300 feet from the breaker box. Rented a ride on trencher cause I had to go through MANY tree roots on the way. Ran 12/3 underground cable on a 20A breaker. NO problems. The breaker has NEVER popped.
Since running the circuit, have built a barn on the sight with the same circuit wired in.
This added a small dorm type fridge and 3 florescent fixtures to the mix. Yes, my wife, who's operation this really is, is judicious in using amperage. In other words she doesnt try running the hair dryer and the 3 water warmers at the same time; but everything else, lights, fridge, fence charger, and spot light charger WORK JUST FINE with no breaker blowing............
NOW........if I was to do it again............a 30A with 10-3 would be worth the extra money to be able to run everything........BUT......common sense does prevail.
At the same time, I ran to another barn where I park the Air Stream trailer. At first I used the 20A 12-3; but noticed when I ran the AC or heat pump that the wiring would be warm.....NOT HOT.........but a definate voltage drop.........Redug the trench, put in 10-3 on a 30A breaker.......NO VOLTAGE DROP and NO cable getting warm...........
So..........Again, if I had to rewire the goat barn/shed power feed, which I will one of these days when there is a problem.......I WOULD USE 10-3 on a 30A............God Bless.....Dennis