The first thing to say is that the existence of God is different from the existence of, say, trees or chairs. The existence of a tree is usually so obvious that you don’t need to present arguments — almost nobody would dispute the existence of tree that was right in front of them. God is clearly different from trees in that we cannot perceive him directly. So we may never have the same surety about the existence of God that we might have about the existence of trees or chairs. But that doesn’t mean we cannot reach a firm conclusion about whether God exists or not.
There are numerous arguments proposed in favour of the existence of God, too many to go through here, so I will present a brief summary of the main ideas:
1) The Cosmological Argument
Everything that has a beginning as a cause. So, ultimately, there has to be a First Cause, that had no beginning (i.e. is eternal) and is able to cause everything else. This First Cause couldn’t be matter because matter is inanimate — you need something else to move it. The First Cause could a Mind, because minds can organise and arrange matter. We call that Mind “God”.
2) Design Arguments
Things that appear to be designed are usually designed. There are some exceptions, for instance if there is some organising principle (i.e. rain flowing down a pipe will only flow in one direction due to the limitations of the pipe), however even in these circumstances you have to ask where did that organising principle come from. Design requires a Designer. We call that Designer “God”.
3) Goodness, Beauty, Truth
Concepts like goodness, beauty and truth are things that are very difficult to explain on purely materialistic grounds. There may be an evolutionary advantage in you sometimes doing good things but there is no reason why we should believe some things to be good and others to be bad. Morality is superfluous to the demands of evolution. If beauty is just a way of finding a mate then it has an evolutionary purpose but why then should we consider the music of Mozart or the art of Michelangelo particularly beautiful? (I am not considering mating with either!) And there seems to be no reason for believing that your beliefs are true beliefs if they are the result of random materialistic causes.
4) Bible, miracles, etc.
The Bible is an amazing book. It contains prophesies that have come true despite being written hundreds of years before the events. The resurrection of Jesus was an amazing event that has strong historical testimony. These kinds of miracles are ways in which God reveals himself to mankind and powerful confirmation of his existence.
5) Personal experience.
A lot of people may start believing in God because of the sort of argument presented above, but will end up believing in God simply because living as a Christian they see God working in their life. As many people look back on their lives they can see providence in things that has happened to them. Now there is no guarantee that everyone will have this kind of experience but it is strong confirmation for those who do.