I think not a good idea to suggest older movies, which don't really age well and are mostly out of context for outsiders. @JohnnyGossamer I think before going into serious Tamil work as @lorddemaxus suggested you should check Bollywood movies, as Tamil movie sort of have regional sensibilities and Bollywood have over the time opened to more universal tones. When you get used to stuff in Bollywood, Tamil movies AND MALAYALAM (the best industry in India) are gonna do well.
Aamir Khan recent films except Thugs of Hindostan are often regarded best by non-Indians. 3 Idiots, Rang De Basanti, PK, Dangal are my picks.
SRK has many romantic movies. The ones I will suggest My Name is Khan, DDLJ (its a classic).
Bajrangi Bhaijaan (2015).
Drishyam, there are multiple of these, I suggest you watch Hindi one from 2015.
Zindagi Na Milegi Dobara and Jodha Akbar from Hrithik.
Few others Andhadhun, Tummbad. Try Pushpa: The Rise.