According to the Wiki, the objectives are as follows:but it depends what achievements are required in each age, and I'm not familiar with the later ones.
Age of Discovery: Discover America (self-explanatory), Developed Nation (As a country not in Europe, Asia or Africa, have 100 development), Control Centers of Trade (Own and control 5 core provinces with level 2 or higher centers of trade), A Large City (own and control a core province with at least 30 development), Embrace Renaissance (self-explanatory, except you need it in all state provinces as well), Hold Three Thrones (Have 2 PUs as a Christian nation), Feudal Society (As a non-Christian nation, have 5 vassals), Present on Two Continents (self-explanatory) and Humiliate Rival (self-explanatory)
Age of Reformation: Join the Reformation (self-explanatory, but includes Hussite), Humanist or Religious (complete one of the two idea groups), Convert Provinces (10, conversions during the Age of Discovery count), Create a Colonial Empire (5 colonial nations as vassals), Converted another Nation (Any way, doing it in the Age of Discovery counts), Asian Trade (Gain the trade bonus in Silk, Spices, Chinaware or Cloves) and Unify Culture (Own, or have a subject own, all provinces of your culture group)
Age of Absolutism: Three Trade Companies (self-explanatory), Multiple Universities (own and control at least 5), Potential for a Large Army (Reach a force limit of at least 200 regiments), Absolutism (At least 90), Emperor of China (have at least 50 mandate), Multicultural Nation (At least 5 different promoted cultures), Win Religious War (Don't have to lead the league, just be part of the winning side)
Age of Revolutions: Parliament (Have a government with a Parliament), Emperor (Have either Empire rank, Emperor of HRE or Emperor of China), Powerful Subject (Have a subject with at least 250 development), High Discipline (at least 125%), Large Capital (at least 50 development), Great Leader (3-star general or admiral) and Control Institutions (Own and control at least 2 origins of institutions).
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