I agree with Graybeard about the .22 and most everything else (Deer wise, at least) I'm sure if I tried I could find things to disagree about. (Why bother, he's giving great advise)
I'm a retired LEO. I started with a .22, then I shot .38/357 for many years as duty rounds.
(.38 with 158gr will take a deer, so will .40 S&W, .45ACP ect... These rounds in the right place will do the job. In the wrong place they are inadequate.)
For the past 20 yrs I've hunted with .44mag & .45 Colts, I don't load heavy/hot loads... I just kill deer and hogs with them. (I also carry one of these year round, it's over-kill on snakes and coyotes, but great practice, and a major confidence builder)