"Near the Medical Center" can put you in a quite good neighborhood, or quite bad. The cities of West University, Southside Place, and Bellaire are better-policed than Houston, which is good if you want a safe neighborhood, and bad if you drive with a lead foot. The areas inside Loop 610, and some distance outside 610, and east of Highway 288 are largely still quite marginal neighborhoods, with some limited nice new expensive housing, but still with some of the Third Ward character and characters. Be careful, some folks from out of state will buy housing in marginal neighborhoods, without enough research, and then be quite distressed to find they are in the same block as a crack house. The Fourth and Third wards are indeed gentrifying quite fast, but marginal neighborhoods do exist within them. If I can be of any help about a specific address, let me know.
I live in Bellaire, and drive to American Shooting Centers to shoot outdoors. I shoot pistols indoors much more often, at Marksman Indoor Range on Nebraska Street in South Houston, and sometimes at Top Gun on Beverly Hill in Houston.
I recently heard about a range somewhere south of Houston, which might be a more pleasant drive than going all the way to American Shooting Centers. I will try to find out more information in the next couple of weeks.
I don't know of any Bullseye leagues in the area, but then, I have not asked around.