You can't "prove" anything with the standard methods of proof in metaphysics, it just doesn't work that way. In other words, you can't scientifically prove something that does not belong in that kind of science. However, there are logical and reasonable arguments in favor of the existence of God.
The First Way: Argument from Motion
Our senses prove that some things are in motion.
Things move when potential motion becomes actual motion.
Only an actual motion can convert a potential motion into an actual motion.
Nothing can be at once in both actuality and potentiality in the same respect (i.e., if both actual and potential, it is actual in one respect and potential in another).
Therefore nothing can move itself.
Therefore each thing in motion is moved by something else.
The sequence of motion cannot extend ad infinitum.
Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God.
The Second Way: Argument from Efficient Causes
We perceive a series of efficient causes of things in the world.
Nothing exists prior to itself.
Therefore nothing is the efficient cause of itself.
If a previous efficient cause does not exist, neither does the thing that results.
Therefore if the first thing in a series does not exist, nothing in the series exists.
The series of efficient causes cannot extend ad infinitum into the past, for then there would be no things existing now.
Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.
The Third Way: Argument from Possibility and Necessity (Reductio argument)
We find in nature things that are possible to be and not to be, that come into being and go out of being i.e., contingent beings.
Assume that every being is a contingent being.
For each contingent being, there is a time it does not exist.
Therefore it is impossible for these always to exist.
Therefore there could have been a time when no things existed.
Therefore at that time there would have been nothing to bring the currently existing contingent beings into existence.
Therefore, nothing would be in existence now.
We have reached an absurd result from assuming that every being is a contingent being.
Therefore not every being is a contingent being.
Therefore some being exists of its own necessity, and does not receive its existence from another being, but rather causes them. This all men speak of as God.
The Fourth Way: Argument from Gradation of Being
There is a gradation to be found in things: some are better or worse than others.
Predications of degree require reference to the “uttermost” case (e.g., a thing is said to be hotter according as it more nearly resembles that which is hottest).
The maximum in any genus is the cause of all in that genus.
Therefore there must also be something which is to all beings the cause of their being, goodness, and every other perfection; and this we call God.
The Fifth Way: Argument from Design
We see that natural bodies work toward some goal, and do not do so by chance.
Most natural things lack knowledge.
But as an arrow reaches its target because it is directed by an archer, what lacks intelligence achieves goals by being directed by something intelligence.
Therefore some intelligent being exists by whom all natural things are directed to their end; and this being we call God.