Use conduit above ground. If its down low it may be considered subject to damage. If higher up exposed UF may be ok but conduit will look better, so you may as well use conduit the whole way. If you use thinwall metal conduit for the above grade parts (thinwall can't be used underground) you need to use metal boxes to ground it. You can use electrical PVC for the above ground portion, since you probably will be using this for the below ground portion you may as well use the same for above ground. If you use UF underground you need conduit to provide protection where it emerges from the ground. Most codes will allow the heavier schedule PVC. Use individual THWN wires in the conduit. UF in a complete conduit run is not recommended, and as said regular Romex is prohibited in wet locations even if in conduit. Obviously the receptacles need GFCI protection, if you provide GFCI protection for the whole run at the source (before the wiring goes underground) you may be able to get a reduced burial depth but its a good idea to bury it the normal minimum depth. Check local codes, some don't allow UF. Most will allow PVC, some require rigid heavy wall steel.
If you use conduit you have flexibility since you can pull multiple wires, you can have one hot for always live receptacles, one hot for switched at the house receptacles, and even more hots for switched lights (even 3-ways). between the house and the structure. If you are using the shed as a shop and need more circuits you need a sub panel, but I assume the power to the shed and the pergola are all going to be on one circuit.