That's a pretty general request. You can get into a pump, auto loader, or a medium priced SxS or O/U shotgun with a $1000 budget. I usually use a pump gun for brush busting bunny's, nothing too fancy as you stand a good chance of scuffing and scarring up wood in heavy brush. If behind beagles, a light toting SxS or O/U shotgun.
I prefer #5 bird shot, just 7/8oz loads for rabbits, less shot to pick out of the meat, good to about 50 yards with a mod choke.
I have shot a fair number of bunnies with a single shot 20 gauge that cost less than a $100. There is not going to be a best gun for what you plan to do, just an optimum one for your type of hunting.
And welcome to The High Road.