Colonization I

Tornadoes have less probability to destroy buildings (suspect %20 from developer notes)
-10% Construction Costs

Colonization II

New units are spawned more frequently (+2 immigration bonus from developer notes)
Droughts no longer affect colonies

Colonization III

-20% Construction Cost
-50% tools and build time for engineers (from developer notes)

Exploration I -

Better chance of positive outcome when exploring ruins
+2 supplies for all units

Exploration II -

+50% more gold from ruins
Naturalists can reveal new bonus resources in mountains and hills

Exploration III -

+1 spotting range for all units
+4 supplies for all units

Army -

+50% attack for mounted units
+50% defense for units armed with muskets

Army -

Armed units can no longer lose muskets in combat
+50% attack for cannons and mortars (from developer notes)

Army -

Units that would die in combat survive with 1 strength
Soldiers can spawn on the docks

Navy I -

New ship types appear earlier
+50% ships defense

Navy II -

Higher chance of evading enemy ships
-10% ship construction cost (from developer notes)

Navy III -

+1 ships movement
Higher chance of capturing enemy ships (from developer notes)

Exploration is a good candidate for early game research investment as getting the 'Better chance of positive outcome when exploring ruins' from Exploration I and '+50% more gold from ruins' from Exploration II will help with ruin exploration; though this is a bit of a gamble as '+2 supplies for all units' from Exploration I is really bad. '+1 spotting range for all units' from Exploration III is also good if you plan on trying to map out the world for the World Discoverer achievement.

Navy is a good alternative candidate for initial investment as 'New ship types appear earlier' from Navy I will likely have good long term benefit. Navy tech is also really good for traders as '+50% ships defense' as Navy I, 'Higher chance of evading enemy ships' at Navy II, and '+1 ships movement ' from Navy III will all help in getting your trade ships too and from their destinations. Your Navy is likely your go to defense against your Monarch when you declare independence so it is useful for that purpose too.

Colonization is a decent mid game investment for research as '-10% Construction Costs' from Colonization I and '-20% Construction Cost ' from Colonization III really help you when you start using Tools to build structures and 'tornadoes have less probability to destroy buildings' from Colonization I is useful in keeping these structures around. 'New units are spawned more frequently' from Colonization II is also nice, though 'Droughts no longer affect colonies' from Colonization is a bit lacklustre.

Army is your long term planning for research. The increased combat values for '+50% attack for mounted units' and '+50% defense for units armed with muskets' from Army I will help you with surviving against attacks from foreign powers (or in attacking them) and 'Armed units can no longer lose muskets in combat' from Army II and 'Units that would die in combat survive with 1 strength ' from Army III will help mitigate attrition. Having Soldiers spawn on the docks is a decent boon too in building up military strength. It should go without saying that Army tech can be useful against the Monarch as well - assuming you don't destroy them with your Navy.