Personal Priority Order:
* There is a lot of jumping back and forth but in general this is going to be your priority order
Absolute 1st Priority:
Although seemingly expensive both in gold and time the best place to start your research is the Construction and Typography Technologies. These techs decrease the time required to finish buildings and technologies which can knock weeks to months off a single build. Try to keep these techs as high of level as your academy level will allow to maximize their benefit.
Your next stop in the Academy should be the Military Tree. Debatably the most important techs in the game are found here and those are the Techs that increase the Attack, Defense, and Hit Points of your troops. I cannot stress enough how important these techs are. They make huge differences in the number of wounded when hitting bosses and even can decide early game PvP battles. You want to start by focusing the Techs that align with the troop types you are going to focus but eventually you will want to max them all.