What size wire should I use for my yard shed, and how deep should I bury it?
One trench the whole way will be easier to pull cable through.  You could make it direct, one straight line, or closer to it than otherwise.

Fences can get blown over, run into, replaced for service, or removed for aesthetic reasons, which will become more of a pain to deal with if you run conduit along the fence.  Also its less pleasurable to look at conduit than to not.

Your municipality or county should be able to tell you your local minimum depth for buried cable.  It depends on your jurisdiction if you're asking about code as I assume.

I would recommend oversizing the conduit significantly to make pulling easier and allow for easy upgrades in the future.  2" can't be that much more than 1".  It will be more durable and it could save a lot of work later on.

You can save costs on grounding by using bare wire, or using metal conduit as your grounding conductor.  But I would not bury metal conduit to avoid dealing with corrosion.  Even a bare ground conductor inside a plastic conduit is going to fare worse than a sheathed over time.

Paint a white line where you plan to excavate and call your utility locating service out to mark obstacles. The white line or shape will keep them from needing to mark unnecessary things in far off parts of your yard.



Rent a trencher and make the work easy on yourself. 



Bury conduit larger than you need, because excavation sucks.

Dig depth is often 2foot minimum for nonmetallic, 6 inch minimum for metallic, but ask your codes department what they will approve.

For wire gaguge, use a calculator like http://wiresizecalculator.net/.



By my estimate you would want 6awg, but maybe the direct patch can cut down distance and let you use smaller.