Academy

It's crucial to know which technologies are the most important ones. All according to your city specialization. Also, you must have anchors if you're a sniper, but this is more advanced. For now,  I'll focus on the basics. A means "I must have", B means "I'll research this if I have enough points or if applies to this specific city", C means "I'll never research this". You can find more info in City Specialization.

Slingers - B - research them, if you're going to use them;
Archers - B- research them, if it's a support city;
City Guard - A - It makes your milita stronger. Behind a wall 25 plus Tower in a revolt world, you gain a lot of free battle points;
Hoplites - B- research them, if it's a support city, or if it's a hoplite city or a conquest city;
Meteorology - B - research it, if it's a city of flyers(Harpies, Manticores and Griffin);
Espionage - B - not a priority, but if you have enough points, you can research this. It will make your espionage cheaper;
Villagers' loyalty - B - Only if you have farming villages in the same island of your city;
Ceramics - A - This is basicly a cheap upgrade in the Warehouse;
Horsemen - B - research them, if you're going to use them;
Architecture - A -  You will economize many resources;
Trainer - B - Essential for OLU, DLU and MOU cities. It makes your barracks recruit faster;
Biremes - B - Main firepower of birs and mixed cities;
Crane - B - Good in your first city or in cities that you have many resources;
Shipwright - B - Essential in naval cities;
Colony Ships - B - Essential in OLU and conquest cities;
Chariots - B - research them if it's a conquest city;
Light Ships - B - Essential in OLU and Light Ship cities;
Conscription - B - Essential for OLU, DLU and MOU cities. It reduces the cost of the troops;
Fire Ships - C - Never build in large scales. Only useful as an anchor;
Catapults - B - Essential in OLU cities in revolt worlds. But very good to keep in all your OLU cities;
Cryptography - C - Waste of research point and resources;
Triremes - B - Essential in Conquest cities;
Phalanx - B - Good in your OLU and DLU cities;
Breakthrough - B - Good in OLU cities. There is more info here, but if you don't know how to use than don't research it;
Mathematics - B - Good in naval cities. It reduces the costs of naval units;
Battering ram - B -
Cartography - B -
Conquest - B -  Essential in OLU and conquest cities;
Stone Hail - C - Why are you going to destroy the city that you're going to conquest? Your catapults already take down the wall, so why keep destroying?
Temple looting - B - Essential in MOU cities. Now, you can plunder favor from inactives or from the enemies and use it to make your mythical nuke;
Divine selection - B - Essential in MOU cities. Phalanx version of mythical units;
Battle experience - C - Waste of resources, research points and poputaion;
Strong wine - C - Waste of resources, research points and poputaion;
Set sail - C - Waste of resources, research points and poputaion.

Notice that most of the technologies are tagged with B, then I recommend you to read the Intermediate Guides to have a better understand to why research this and not this, and why upgrade in this way and not in this way. To find out about anchors check Advanced Guides.