Walking in woods near towns and cities you're more likely, I think, to be threatened by a pack of feral dogs. I have no qualms about shooting them if they're aggressive towards people.
Most wild critters are wary of humans and will do the same things we do. Keep a watch on the target and avoid approaching. Only when they start closing in should you start getting worried. I've had a mt. lion watch me from less than 50 yards, but I think s/he was more curious than anything. Or waiting for me to leave before crossing the road I was on.
The one rule of thumb I have is that if I come across the young of a predator (bear, mt. lion, wolf, even pigs) then I will immediately backtrack to avoid meeting mama.
BillCA in CA (Unfortunately)