This topic has been pretty well-discussed here:
https://www.secretprojects.co.uk/threads/what-if-iran-never-had-a-revolution.37448/
I suspect you would have seen the 4 Kidds, up to 8 Kortenaers, and 6 Type 209 SSKs in the 1980s, plus light craft. Iran's main focus is always going to be the Army and Air Force, so the Navy becomes an afterthought. Beyond that, it depends on what path Reza II takes, because the Shah would almost certainly have passed by 1982 even without the revolution.
Investment in the Navy is going to depend on two factors: the land threat posed by the Soviets and the Iraqis and the air/sea threat posed by the Saudis and Arabs. The Saudis and Iranians were on the same side of the Cold War, but that really only papered over their rivalry, so it will become the driving factor in the region after 1991.
One thing I would expect is a larger amphibious capability to threaten or protect the Gulf islands and sustain Iran's overseas peacekeeping forces in Africa and Lebanon. The Iranians bought a number of Yarrow landing ships in the 1970s, and I would expect something like a Round Table class LST, almost certainly from Britain, in the 1980s.
With combatants built in the mid 1970s to 1980s, the Iranians would probably start looking for replacements in the 2000s. I don't think they would try a DDG/FFG split, but the natural direct replacement for the Kidds would be a Burke. I think an all-frigate force would be most likely. Potential choices there include the MEKO A200 like what the South Africans bought, a derivative of the German F123 and F124 designs, the Spanish F100 design, or an air defense-focused La Fayette variant like the Saudi or Singaporean ships.
The desire for an aircraft carrier of some sort for regional power projection would be natural given the Iranian position, but I don't think it would be viable. An aviation-capable amphibious assault ship in the 1990s or 2000s would be much more likely. Given the Iranian predilection for British amphibious and logistics ships, I would expect a copy of HMS Ocean if the purchase is in the 1990s. In the 2000s, Mistral would probably be more likely, and the relatively low cost could allow the purchase of two or three. Alternatively, the Spanish BPE (Juan Carlos I) would offer the amphibious capability as well as Harrier capability, though that might be aspirational for a relatively long period.