For a .22 auto handgun, I'll chime in with praise for the Ruger. It was Ruger's first gun & has been made for decades(over 60 yrs now?) beginning with the MK I. I have a MK II Target, blue steel, 6 7/8" heavy tapered barrel, little used & still shoots much better than I ever could or will. With the Rugers, make sure spare magazines are genuine made-by-Ruger, clean it once in a while, & you'll find it extremely reliable. There are *lots* of good ones out there if you decide to buy used.
I have no experience with the Browning, but it seems to be very popular also & should be a safe bet.
.22 revolver: in single actions, the Ruger Single Six is the way to go. The blue steel versions are lighter than stainless, *Because*: Stainless Single Sixes are all stainless steel. Blue Single Sixes have Aluminum grip frames & ejector rod housings. You could look up the weight difference(not huge) at Ruger's website. My 6 1/2" barrel Stainless Super Single Six is a great .22 six-shooter.
Double action revolvers: good used(& often long discontinued) models abound, but you gotta know how to separate the wheat from the chaff. I have a S&W Model 63 Stainless Kit Gun- it's definitely in the wheat category!
Of those currently made, S&W& Taurus are certainly worth a look. In used /discontinued, other names to look for are H&R(Harrington & Richardson), High Standard, Charter- and of course, Colt($$$$).
Center fire- with today's ammo prices I'd go with a 9mm automatic. Choices are mind boggling.
Whatever you get, let us know, will ya?
Safe shooting!