I did a similar project last year - ran about 150' of direct burial (4/0 4/0 2/0) cable from pole to barn, and 350' of 3/4" pex from meter to barn. I'm in central Texas, so no big worry on frost line, but still tried to stay around 3' down on water and a bit deeper on power.
Quick suggestion to OP, if you are going to rent, spend the bit of extra money and get a tractor with trencher as opposed to a walk behind. I wasted a day's rental and a lot of effort on trenching my clay soil and never really got the depth I needed. Traded up on the rental and completed the entire job in less than 2 hours using the proper equipment. The trench was much deeper, cleaner, straighter, and not filled with my sweat. Also, as someone else suggested, place hazard tape about 6 - 10" below grade. I just used caution tape, but now see that some has a metal strip for easier detection. Wish I would have known or thought about this.
Not sure if we are necessarily to code, but think did a good job. Have a 300 amp box at pole, grounded, and then pulled into 200 amp box in barn. That is way more than I need, but my dad had spare boxes along with the breakers, so we went with it. We wired 6 receptacles, and 2 switched overhead floods, plus porch light, and 30 amp outlet for travel trailer.
The water was pretty easy - ran the line to foundation of barn into a 3/4 x 3/4 x 1/2" T - I then plumbed a frost free hydrant to the 3/4" and ran 1/2" along foundation and then into spigot inside barn.