So do dogs, cats, horses and other pets go to Heaven?
While the Bible doesn't come straight out and say "yes," it does imply through its discussion of the redemption of God's creation that our pets will greet us in the new Heaven and new Earth. We can see here in Romans 8 that even creation has been in bondage and decay because of human sin and is groaning for liberation that the final restoration will give. This passage shows that God is planning on restoring His creation, and animals are a part of that creation. Romans 8:18-22 which describes creation groaning for redemption.
"I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. For the creation waits in eager expectation for the children of God to be revealed. For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the freedom and glory of the children of God.
We know that the whole creation has been groaning as in the pains of childbirth right up to the present time." (NIV)