Everyone is advising against using AL Service Entrance cable. That's what you want, because it's a service entrance. Absolutely nothing wrong with feeding a panel with AL wire. It's safe, done all the time, and cheaper. Now your branch circuits are another matter. But to feed the panel, use AL. I bet you a donut if you pull the cover of the house panel, it's fed with AL wire. That's all the power companies use. If you feed the panel in the garage out of your main panel, you want to be sure the breaker is rated for AL wire and use this stuff:
Trenches, in MD where I am, we have to go 36" deep for water lines to keep them from freezing. It's common practice (and code approved) to dig a 36" trench, lay your water line, backfill to 12-18" deep, then lay your conduit for electric service.
Speaking of conduit, conduit is like garages. Figure out the size you think you need, then make it bigger. I would go 3" from the house to the shed for the heavy wire. Why? Ever try to pull heavy wire? It's a witch. The bigger the conduit the better. And while the trench is open, throw in a spare 2" conduit. It's cheap. And then when you want a phone, or security system, or cable TV or whatever in the shop, the conduit is already in place.
As far as wire size goes, there are calculators all over the web that will calculate it for you. You may need 3 conductors, may need 4. Depends on your locale and what version of the NEC they use and enforcement. Best to ask a local inspector because the answers are different all over the country. Ask them what they want to see, then give it to them and your inspection will go A LOT easier. You are pulling a permit and getting an inspection I assume.