12 gauge will do nicely. 14 would be fine even. I'd go 12. Heaver wire is always a good ideal. 8 gauge, would be around 40 amps. Over kill for lights and a few outlets.
Unless you are running conduit, you need something with a sheath rated for direct burial. To do it properly, you really need to put a small subpanel in the pole barn, they are cheap at lowes. Code will also probably want a new ground out there as well. I would run a 30 or 40 amp service and then wire the barn with Romex.I am going to run about 150 feet of buried electric wire to a small pole barn. I will have 3 or 4 lights, and about 6 110 volt receptacles. No heavy duty work on the outlets, will use mainly for lighting. What size and type wire would you think I need to use to feed the building?