The Iran-Iraq war was the first one, with Iraq funded and supported by the Gulf states.
Supporting Hamas and Hezbollah is strategic in this context. The Saudi regime wants rapprochement with Israel and to remain aligned with US interests. But neither of these are remotely popular in the Saudi population. By funding guerrilla warfare against Israel, Iran and to a lesser extent Qatar, keeps the Sauds discredited and unpopular among at home and in other Arab countries. The same applies to Egypt, another regional rival of Iran, whose government have never been off the defensive with the Egyptian people and wider Arab opinion since normalisation with Israel.
Obviously Hamas and Hezbollah themselves are only interested in fighting Israel and not the wider regional conflicts. But Iran itself uses that conflict, quite cynically, for wider geopolitical goals. Its stance is the reason that, from Afghanistan to Turkey to Tunisia, it can always find allies who want to challenge the Gulf states vision for the Middle East. Iran supplies the weapons and the know how, but there's never a shortage of locals to drive the car bombs.
There is an interpretation of Iran's behaviour which sees it as a source of Muslim pride for standing up to imperialism, and suggests in contrast that the Saudi leaders are too decadent, too corrupt, and bring shame by ignoring injustice and exploitation done to Arabs. I would certainly question this, but it's not an unpopular discourse in Saudi and other Arab countries.
If you have never come across the idea of the conflicts in Syria, Lebanon, Iraq, Yemen and Palestine etc being part of a long game of proxy war and influence between Iran and Saudi, I would question how broad your sources of analysis are.
