I live on the westside of Indianapolis. I'm looking for some good public areas to hunt deer and coyote. I'm not really sure what's decently close, or worth the trip anyways.
There is also public hunting ground on the south side of Raccoon Lake which wouldn't be terribly far for you. Never deer or coyote hunted there but its my go to squirrel place.
Owen Putnam State Forest. About 8000 acres or so of public land. Its between Cloverdale and Spencer. Raccoon is decent hunting but you may as well forget it during firearms season. Archery or muzzleloader are your best bet there.
Clark State Forest down around Henryville is an awesome spot if you're willing to do the homework. You will have to do the homework though, and the leg work. There's some punishing terrain in that area and the wildlife takes full advantage of it. Coyotes are sparse, in my experience, but deer, turkey, and squirrels are abundant. I've taken a number of limits of squirrels out of there, most in close proximity to parking areas.