The wolf shall dwell with the lamb,
and the leopard shall lie down with the young goat,
and the calf and the lion and the fattened calf together;
and a little child shall lead them.
In Genesis 1:25, God created animals and said that they were good. In other early sections of Genesis, both humans and animals are said to have the "breath of life." Man was given dominion over every living thing on earth and in the sea, which is no small responsibility. We understand that the difference between man and animal is that people are made in the image of God, according to Genesis 1:26. We have souls and a spiritual nature that will continue on after our bodies have passed away. It's difficult to make a clear case that our pets will be waiting for us in heaven given the silence of scripture on the matter.
However, we do know, thanks to two verses in Isaiah, 11:6 and 65:25, that there will be animals living in perfect harmony in the millennial reign of Christ. And because many things on earth seem to be a shadow of the wonderful reality of heaven we see in Revelation, I have to say that our relationships with animals in our lives now must be preparing us for something similar and good to come.