I go hiking a lot and I noticed that if you just ignore the animals they will be much more comfortable with you
Walk by deer, take a look, and keep walking. It won't run away and will continue grazing. Same thing with birds, just pretend they're not there and they won't fly away when you walk by.
One time saw a black bear a couple feet off the trail. We locked eyes for a few seconds and then it went back to doing whatever it was doing before we met.