I recently purchased a house that has two 30 amp breakers going to a shed for lights and outlets. How would I go about doing that correctly? Can I tie the two 30 amp breakers to the shed together and install a 60 amp subpanel in the shed? Then have 15 and 20 amp breakers for the lights and outlets? Any advice would be appreciated.
What you have is NOT legal and really should be corrected. You CANNOT have general use lighting and receptacles on 30A circuits, even with #10 wire. You also do need a means of disconnect at a detached structure.
You can use this feed to power a 30A-120/240V sub-panel using a tied two-pole 30A breaker in the main panel. This is only true if there are two hot wires, one insulated white and a bare or green ground. You CANNOT do this with only a 3-wire feeder. You can use a small 100A main breaker panel being fed with the 30A feeder. The 100A main will only serve as the means of disconnect and the 30A breaker will protect the feeder wire.
NO, NO, NO. The size of the breaker is matched to the size of the wire it is connected onto. I will assume that the two 30 amp breakers are in tandem and thus to two phases of your panel thus they represent 30 amps on two legs for a 220 V circuit. Thus - 30 amp max 220V or 60 Amps TOTAL amps at 110V (30 on each leg). Usually this type of setup would be run with two 'hot' legs, one neutral and one ground. Even if you are only using 110V appliances, tools etc. it is important to remember that those two legs must be connected to different phases in the panel. If you are unsure - consult a licensed electrician.