common argument for existence: the first cause argument. Basically it says, "the world exists and things happen, how could it happen without a cause? since all things have a cause, there must be a "first" cause, the prime mover, aka God"
this argument sidesteps the notion of "what caused God", to which a common reply is, "nothing, God is by definition the first cause"
there are still others, such as Anselm's Ontological argument, which basically says (IIRC) that all things have a hierarchy of "being". There must be a thing with the "highest being" so to speak, and that thing is by definition God.
then there is the argument: God exists because the Bible says so.
that's sort of circular because the Bible was "written/inspired by God"
the most common argument for non-existence is the problem of evil: if God is good, how can evil exist?
This is more an argument about the nature of God, that is, God being good. Notice that the arguments for existence primarily focused on the idea of "God" as a "first cause," without going into detail about the nature of God as a being.
hope that gives you some intro for further analysis and research.
actually, the issue about the problem of evil is less about the non-existence of God, but it is sometimes used as an argument for that position...