Just a couple random comments:
For what you're doing, I'd say use either 5" diameter cedar fenceposts (round, untreated, bark removed) (4" diameter won't last as long) or pressure treated 4x4s. Either will be FINE for you, longevity wise.
I would recommend closer to the 3' suggestion than the 2' in terms of how deep to set the posts. Probably I'm biased 'cuz I'm used to fenceposts having to keep horses in, but still, it gives you a stronger post.
Personally I would not use cement as it makes them a bugger to remove in the future, and as this is a residential backyard not a farm, it is quite possible some poor soul will want to remove them someday
A gas-powered post hole digger will save you time, but IME it will NOT save you labor. It will give you
quite the wrestling match and possibly some interesting bruises
Just to highlight 2 things that others have said, because they are of crucial importance: don't rush thru the tamping stage. Tamp every 4-6" of fill and tamp evenly and HARD. And, the advice to put a couple big flat rocks on the sides of the posts (down in the hole) makes a GREAT BIG difference to stability. It's especially important if you're cheating on the depth of the hole. But you must be careful to tamp the dirt well under and around them as well as over them.
Oh - and make sure to lay out a string, stretched tight between stakes, that marks where you are setting your posts. Even a slightly wibbly layout can make it significantly annoying to try to attach the rest of the lumber for the run, plus the way it looks will annoy you for as long as you live there
Have fun,
Pat, who really needs to straighten (meaning, reset) the leany posts of about 60' of horse fence this summer, but has a bum shoulder at the moment that is providing a good excuse