Both of the above speak wisely.
.38 are cheep to buy, .357 will give you plenty of boom to get your grin on. .45LC is more authentic, and can be loaded to have a bigger boom.
I load my .45LC pistol rounds with black powder. As a slow burning powder, the recoil is more of a "thump" than a "snap", and nobody I've let shoot it has had a problem with the recoil (though I do start the ladies/smaller guys off with cowboy loads to get their confidence up). Plus there's pile of smoke. Very John Wayne
If you're doing this as baby-steps to get you into cowboy action, you might want to also consider what you want for a rifle. Matching caliber will make your life easier
Personally, I went with .45LC partly because a little bit more modern rifle (in my case, a Marlin '94) will have the look for cowboy action, and can be loaded heavy for perfectly good bush gun (shorter range, heavy bullet to push its way through the twigs).
Really, you can't go wrong with either