I have had to say goodbye to seven beloved dogs over the years. It is an extremely painful experience. What makes it even worse are the well-meaning, but foolish condolences regarding how the animal is waiting for you on the other side of "Rainbow Bridge", a complete fabrication made up in the 80's.
I have a wonderful old rescue dog now, that I pray to God will last a while longer. With each of my dogs I was thankful for the time I had with them, knowing it would not be forever. The pleasure of knowing them outweighed the pain of their loss by an infinite about, and I would eagerly sign up for another chance with them again, if that were possible.
The Bible doesn't tell us what happens to animals after they die, but they are in God's hands, and that is good enough for me. I'll find out soon enough.