Which gun is small, shoots fast and also isn’t that small of a bullet

I’m looking for something handheld. I don’t know anything about guns at all. Also not trying to spend more than I have to on it. Also how do you buy a gun in Florida do u just go to a gun shop and start there? Also does Florida do actual mental health checks where they look at previous mental health diagnoses and look into it on their own, like I’ve heard other states do, or no? And if anyone knows where private sales happen (in tampa) so I could allegedly get one w/o going through that whole process lmk. Thank you.