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Agriculture, lumbering, masonry and mining give you bonuses to your base resource production, 10% for each level, and are pretty cheap and quick to research. Informatics is used to scout flats, NPC villages and other players so that is very important too. Archery gives you a bonus to your archers range, with matters a lot in outcomes of battles where lots of archers are included, so as soon as you start getting a few thousand archers or more it's important that you try to get the archery technology as high as possible.

You're not going to max the important technologies right away, but try keeping them a few levels ahead of the others. Getting informatics and archery to level 10 should be done pretty quickly though, since it will help you a lot especially when farming NPCs, and NPC farming is a huge part of Evony. The other important technologies in my opinion are medicine, horseback riding, compass and military tradition. Military tradition increases your troop training speed, medicine saves some troops from dying in combat, horseback riding increases mounted speed and compass increases foot soldier speed. Overall the combat technologies are the most important, the reason you also research bonu production technologies is cause they are very cheap and fast to research. City technologies are not that important at all, but with time all of your technologies should reach level 10.

Here is a list of the various technologies in the game.

Resource bonus Technologies.
Agriculture.
Lumbering.
Masonry.
Mining.

Combat Technologies.
Archery.
Compass.
Horseback Riding.
Informatics.
Iron Working.
Medicine.
Metal Casting.
Military Science.
Military Tradition.
Privateering.

City Technologies.
Construction.
Engineering.
Logistics.
Machinery.
Stockpile.