What is the best way to prove the existence of a necessary being as opposed to an infinite series of contingent beings?

These may be of some value -

Avicenna’s argument from contingency, Part I

Avicenna’s argument from contingency, Part II

Classical theists like Aquinas didn't actually argue that the universe had a beginning, to quote Edward Feser in The Last Superstition -

Aquinas does not argue from the claim that “everything has a cause,” nor, contrary to what Dawkins thinks, does he argue that the universe had a beginning and that God must have been the cause of that beginning. His aim is to show that given that there are in fact some causes of various sorts, the nature of cause and effect entails that God is necessary as an uncaused cause of the universe even if we assume that the universe has always existed and thus had no beginning. The argument is not that the world wouldn’t have got started if God hadn’t knocked down the first domino at some point in the distant past; it is that it wouldn’t exist here and now, or undergo change or exhibit final causes here and now unless God were here and now, and at every moment, sustaining it in being, change, and goal-directedness.

There is also r/classicaltheists if you're interested.

/r/Catholicism Thread