You should finish the floor with tile or carpeting once you check to make sure it's structurally sound. Insulate the walls and put drywall over the walls and ceiling. Insulate the ceiling from the cold and hot weather but don't place the insulation too close to the sill plate. It needs to be open a couple of inches in order for it to breathe from the soffit vents (if any) to the ridge vent (if any).
Pre-wire outlets and lighting where you think they are most needed and install electric baseboard. There are some good electric baseboard lengths that don't need a furnace but instead rely on a thermostat either installed somewhere in the room or sitting right on top of the baseboard. I'm not sure how extensive of an answer you are looking for because you state that you want to convert the shed into a bedroom then you say you want to make a studio apartment.
You will need plumbing such as water piping and drainage if it's going to be more than just a bedroom. Let me know if I gave you enough information or if you need more.
Hope I Helped :-)
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