The Khomeinist regime in Tehran has long preferred not to invade other countries and rather rely on proxies to project Iranian power. Iran first and foremost wants to control the Khomeinist Apocalyptic region, the triangle of Mecca, Jerusalem and Damascus. There can be little doubt that the Iranian military will cross Iraq to reach Syria in order to fight Israel which will defensively invade Lebanon and Syria in order to destroy Hezbollah. This is known in Israel as the Northern War. Once America and Israel or Israel alone destroys the Iranian nuclear weapons program will Hezbollah unleash vast barrages of rockets and missiles against Israel which will invade Lebanon and Syria in order to destroy Hezbollah, establish an Aramean state in parts of Lebanon, install the PYD as the new government in Damascus and establish new international borders of peace through agreements with the new governments in Damascus and East Beirut, including Israeli annexation of Median Jewish (Alawite and Druze) regions.
The outcome is a given, Israel will win and the Iranian axis will be defeated or else Israel will cease to exist. A more interesting question is whether Iran will also invade Saudi Arabia? This is difficult to answer indeed. The main reason Iran wants to conquer Israel is that Khomeinist theologians believe that the Hidden Imam will not appear unless Israel has first been destroyed. From a tactical perspective would it make sense to only fight Israel and possibly the SDF rather than repeating the German mistake of fighting wars on two fronts simultaneously. However, we cannot presume that Tehran makes purely rational military decisions since the obsession with destroying Israel is not rational at all but rather apocalyptic and messianic. In fact, it makes much more strategic sense to conquer the GCC countries with their vast oil resources than to conquer Israel. Then there is the existence of the new Molotov-Ribbentrop pact between Russia and Iran according to which Moscow gets Europe and Tehran gets the MENA region. However, an early attack on the KSA means war with the United States, is Iran ready for that at this point? There is no doubt that Iran intends to conquer the Arab world and so it would certainly make sense to delay war with the GCC until Russia has fully armed Iran. However, we cannot say with certainty what the Iranian war plan is following the destruction of its nuclear weapons program.