I second the comment on the single six as a good practice gun. Makes reloads fast and a matter of habit. I always dump my brass on the ground and hunt it up after I'm done shooting, so as not to get in the habit of holding on to the brass. Bad habit if you're in a hurry.
Using your sixgun for hunting rabbits is a good thing for practice. Shooting some "modern" plate or combat shoots is fun and is good practice, as well as cowboy shoots. Just going out and shooting 500 or a thousand 22's at a time on a nice afternoon is pretty good practice. Long range shooting is also a lot of fun. Even the 22 can do pretty well out to 300 yards if you have dry ground to see your hits. Left handed, one handed, two handed, fast shooting two handed cocking with the left thumb at cans or rocks is good practice.
For games go out with another guy or more and do "follow the leader" with unusual or difficult shots. Dueling trees are good fun. You will have to find your own pace and level, then keep trying to slowly improve it all the time. Just remember, it's easier to get fast AFTER you get GOOD, than it is to get GOOD AFTER you get fast. At a plate shoot, you can beat a LOT of modern shooters if you aim and hit with every shot, instead of trying to be as fast as you can.
If you find a load your gun likes you shouldn't have too much trouble getting a group under 2" at 25 yards, certainly with a rest, and with a lot of practice, offhand reliably.