What should I do if I ever find myself trying to help an injured animal (such as a bird)?

So today, after I got home from church, my dad walked outside and saw an injured bird (it was a female Red Breasted Robin to be exact) in the middle of our street (fortunately, cars never really pass by on our street). It's left leg was injured, and it couldn't really fly.
My dad then grabbed a box, and we gently put the bird in the box. I then called every shelter and refugee center I could find that was located in our area, but most of them were closed. Luckily, I found this one shelter that's in the area, and they said they would take the bird (and they did). I was just in time, too, because they closed at 3 (it was currently 2:15 when I called).
Did I do the right thing? I didn't want to leave the bird to die, but at the same time I'm not sure if it was okay to give an undomesticated bird to a shelter since it is wildlife.

I feel really bad for the poor bird, because it was scared to death. I wanted to hug it to reassure it (even though that probably would have scared the bird even more). T^T
I hope it's doing well. ;-;