It is obvious that we can’t find out whether animals go to heaven just by looking or thinking about it, we have to turn to what God has said in the Bible. However, this is not a question that the Bible addresses directly. From what I understand of those times, animals were for doing work and pets were not a big thing!
What the Bible does tell us that may be relevant includes the following:
1) As far as whether an individual animal “goes to heaven”, the Bible makes no suggestion one way or the other that animals have a “soul” or “spirit” or the like that continues after their death. But the Bible does make it clear that humans have a special relationship with God that the rest of the animal kingdom does not have. Genesis 1:26-28 tells us we are made in God’s image and given a special role in the world. And only humans (of the material beings) are held accountable to God. So it seems most likely that only humans have the prospect of eternal life.
2) As to whether there will be animals in general in heaven, the Bible is again not specific. However, “heaven” is not a disembodied spiritual existence. Rather, it is a “new heavens and a new earth” (Revelation 21:1) in which we have a body “raised in glory [and] power” (1 Corinthians 15:42-44). This new creation will be like the existing creation in some ways, but far surpassing it in others. But exactly how the Bible doesn’t say, because it speaks about the new creation in images not literal descriptions. But we can’t exclude the possibility of animals, even if we can’t be sure what they would be, or what role they would play.
3) Indeed, the Bible speaks of the best of this present order of things being taken up into the new creation. In Revelation 21:26 John says, “The glory and honour of the nations will be taken into it.” Might this include the best of human-animal relations, such as the partnership of a rider and horse?
4) The central thing the Bible says about heaven is that God will live with his people and so there will be no more tears or pain (Revelation 21:3-4). “His people” means those who in this age have put their trust in the Lord Jesus. Whether or not we have animals, the joy that they bring us will be available in some form or another, only far greater.