I've got a decent sized shed out about 150ft from the house, I plan on running the cable underground in conduit. But I am unsure of what style/size cable I need to use for a 50A circuit 220/240V. I'm assuming that I probably need to use service entrance cable and can't use 6G stranded. I've tried contacting the city for this information but there useless.

I think that depends on your municipality. Check to see who inspects new electric installations. I have a 220v panel in my garage and used 8 gauge.

Minimum size copper wire is #6 to serve a 50 amp circuit 100% load, at 150 feet. If you are planning to have motors starting, the distane will cause the motors to start slow and hard, so I would use #4 copper wire.

Copper wire will probably need PVC conduit and THWN insulated wire, 3 -#4 with two black and one white wires and a #6 Green wire for the equipment ground.

Or you can use #2 aluminum URD direct buried cable, just be sure to treat the connections proeprly with ani-oxidant grease.

Conduit needs to be buried 18in deep, and cable needs 24 in deep.

City inspectors do not tell people how to do the wiring, they are not allowed to normally.