Research priorities are interesting in this game because there are few "no brainers" and you can't get everything - there is a genuine choice.

This is the research order I am following in my first proper game:

(1) Get your free unit: From my limited experience, it seems like the first mission you get is to research a military tech relevant to your clain (way of sword for Shimazu, way of spear for Oda) so it probably makes sense to do that first.

(2) Heaven and earth: 50% more arrows IS a no-brainer.

(3) Farming: I then went down the farming improvements up to Chonindo for the faster economic growth from food surpluses.

(4) Naval: I started to get embroiled in various naval engagements and found the tendency of the AI to retreat far from battle frustrating. So I worked down the naval tech tree which is fairly self-contained. I am not sure it was necessary, but after I had the 20% more sea movement, things seemed a lot less frustrating. I should have gone for fire arrows before going down that path though.

(5) Agents: I started specialising provinces for better agents, so focussed on the left chi path. Ultimately picking up Ninjutsu mastery to get Japan's first Infamous Mizu Shobai District.

(6) Calligraphy: I belatedly got the Confucian Academy, benefiting from the combined +30 diplomacy and +30% trade tariffs from Calligraphy and Tea Ceremony.

(7) Master and Legendary dojos: I am now working on the bushido techs. Having one turn recruitment of samurai during realm divide sounds very attractive, as does being the first in Japan to build the legendary dojos (to get the bonuses for those feats).