.177 or.22... you might as well ask "Which came first,the chicken or the egg?", If you want a subject less controversial, then try "Labour or Conservative?", it really can get that ugly when folks post this question. I've seen otherwise sensible men use the most ridiculous arguments both for and against both calibres in the past so lets concern ourselves with what's
really important.
Based on the fact that you are asking about the purchase of what seems to be your first decent (if not your very first 'proper') air rifle, I'd say that by far and away your top priority should be PRACTICING with it. LOTS. SAFELY.
If you have the chance to go out with someone who is an experienced and responsible hunter with the air rifle then do so. (I added 'responsible' because I've seen far too many experienced fools
) Otherwise, find yourself a safe place to shoot, and practice. Then practice some more. Shoot from various positions and at varying ranges and angles until you can be certain of putting every shot in a circle no bigger than 1 1/2" diameter at whatever range you expect to shoot at. Then practice some more until all shots are in a cicle no bigger than 1". If you can't get close enough to put the shot in a 1" circle on your intended quarry, then you need more practice on your field craft
and your marksmanship.
There has been an awful ot written here and elsewhere on the net about effective ranges of air rifles, kill-zones on quarry (the area to aim for, and hit) etc etc so I won't go into that here, but I will say that the best advice I was given when being taught to shoot was this,
"Always remember, no matter if it's a big stag or a ruddy rat, it's just as alive as you are, and if it ain't
"Dead for certain" before you squeeze that trigger, then you've got no right on this earth to squeeze that trigger at all."
Shoot safe, and shoot well, whatever the perishing calibre