I’m very sorry to hear about your dog passing away. I’m glad, though, that something good has come out of it - that you’ve started thinking about God and reading the Bible. That’s great.
In terms of what happens to animals when they pass away, it not the easiest question to answer, because the Bible doesn’t speak directly about it. In biblical times, animals were seen mainly as either wild animals, as being useful for work, or as useful for food. They didn’t really keep pets as companions as we do today. Because of how they saw animals, the question of whether animals go to heaven isn’t really dealt with in the Bible. We, of course, now live in a time when these sorts of questions occur to us, because we have pets who mean a great deal to us.
Since the Bible doesn’t answer the question directly, we need to look at what we know about God, and work from there.
The first thing we see in the bible is that as the Creator, God loves and enjoys all he created. It was good (Genesis 1:31) and He cares and provides for it (Psalm 104).
The Bible also tells us that God is good and that he always does what’s right. Psalm 119 verse 68 is one place where we are told that:
Finally we see in the bible that the thrust of God’s activity in the world is directed at saving humanity from their sin. He has a special plan for humanity that is above and beyond the rest of creation (just as humanity is the crowning achievement of his creative process) and so all he does in creation is directed towards this end. So the untimely death of your dog was part of God’s purpose in helping you to come back to him.
So bringing this all together we can see that while God’s focus is on humanity, he does love and care for all creation and would not do anything cruel just to spite it. We can’t say for sure what happens to animals but we can trust the goodness of God to do what is right.