If I install a 30 amp breaker in my breaker box, can I use 15 amp outlets on the circuit, provided none of the appliances used exceed 15 amps?
This simply will not work.  The only receptacles you are allowed to put on a 30A breaker are 30A receptacles.  Those are unfit for your purpose.  

Further, if your expected loads are near 30A, you should be derating this by 125% (37.5A) and provisioning that much power - I.E. 40A.  And they don't even make 120V/40A receptacles.  

The right way to do this is to fit a sub panel at the shed, and have individual circuit breakers protecting each #14 or #12 circuit.  

I am also concerned about what your feed cable is to the shed.  This too will be a limiting factor in your total possible load.  There are ways around them, but you should not use those ways until you are well past mismatching sockets, wires and breakers.