Rule of thumb for 1st handgun... a .22lr as named above (or others not yet named... like a Ruger Single Six).
Good for target practice, trigger/sight control, safe handling, training w/ others, plinking, small game hunting, could be used for SD. Good luck finding bulk ammo for that right now.
Next up, in way of centerfire handguns, some say 4" revolver in .38 or .357 others say a semi-auto in 9mm. Be kinda hard pressed to make that w/ your budget... maybe some older lightly used S&W 10/64's or a used Ruger 89/95 might be had cheap enough. Theres always HiPoints. Ammo will cost more (when you ever do find any). Far better for SD and hunting larger game. Holes in paper don't care about size (except being larger makes it easier to see them).
So, as the Great Mahoo asks, what's it for? Answer that, you're halfway home.