Okay the two cents of a certified instructor (ME). You have taken the most important first step that you possibly can. There is another step to take and that is not which gun. There are tons of informed opinions here about that. MY suggestion to you is to go to a gun show and....ahem...."fondle" as many as you can. Pick them up, hold them in your hand. Take a sight picture. Imagine carrying it either on your person in a holster or in your purse. Can you access it quickly? Can your trigger finger easily reach the trigger? Ask permission to "dry fire" the gun at the table. Is the trigger firm enough that you KNOW it's being pulled but not so heavy as to disturb the sight picture? Even a .22 can have an excessively heavy "pull" and using a gunsmith to alter the pull weight might not play well in court. Better to find a factory trigger you can use well.
I would recommend nothing smaller than a 9mm in semi auto or a 38sp in a revolver (even .380 is too small IMHO), but YOU must decide. Nobody ever died of a loud noise. You should get the biggest caliber you can shoot and hit with, consistently. Everything, in the final analysis is shot placement anyway. If you must use a .22, you must be able to consistently hit the eyeball under stress. Remember that standard in relation to caliber size. Israeli MOSSAD uses .22's but they hit the eyeball...and it's instantly "lights out."
I let my clients for the Florida concealed weapons permit pick out their gun, then I teach them to use and carry it well. When you find your gun, you'll know it. It will just feel right. Like "Excalibur" fit the boy King. To my wife, a Colt Combat Commander .45 "felt" right. I taught her to use THAT gun. Now she has her very own and loves it. For my daughter it was a Ruger SP101 in .357 magnum with a 3 inch barrel. That was her graduation present a couple of years ago when she got her degree. Now, she's getting married and I have to make sure her husband is as good at handling the gun as she. Choice of weapon and caliber is all secondary. Getting the certified instructor to teach you is the first step. Here is the next:
Join an IDPA (International Defensive Pistol Assoc) shooting club and PRACTICE PRACTICE PRACTICE! This group uses real life street scenarios to instill proper mindset and use of good tactical doctrine. Being able to punch holes in a paper target is not the same as self defense! IDPA will teach you to use cover; draw properly; shoot from uncomfortable and odd positions; shoot and reload on the move....and it will let you do so with safety officers and some very experienced folks (people who have often "been there and done it" for real) helping. AND it's TONS of fun!
I have no idea how the Israeli's assassinate people but my bet would be "back of the head".