Electrical – Should both ground and neutral in an electrical subpanel be bonded to the service ground

The neutral and ground MUST NOT be bonded at a sub-panel. They should only be bonded at the main service panel. If you bond them anywhere other than the main service, the neutral return current now has multiple paths, including though your ground wire.

You should be able to buy a second bar for the sub-panel if it really is meant to be used as a sub-panel. The neutral bar will need to be isolated (it should have plastic insulator separating it from the case). The ground bar should be bonded to the case.

The ground wire from the house must be connected to the sub-panel ground. Since it is a detached building the ground rods are also required by code and should be tied to the sub-panel ground. The ground to the service is the most important because it will provide the low resistance path back to the service in case of a fault.

If you're in an area that has adopted National Electrical Code, you'll have to run a 4 wire feeder. You'll also still need the ground rods at the shed, which you'll bond the grounding bar in the panel to.

If it's an existing 3 wire feeder, and there are no other conductive paths between the buildings. Then yes, you'd bond the grounded (neutral) bar. However, if there are other conductive paths between the buildings (water pipe, conduit, gas pipe, etc.), then you'll need a 4 wire feeder.

tl;dr

If this is a new installation, you'll need a 4 wire feeder.