Infantry can be trained in the barracks. The higher its level the faster the troops are trained. If you want to build more advanced troops than the first troops, you’ll need to research them using the academy.
|Training costs for Roman troops|
|Training costs for Teutonic troops|
|Training costs for Gallic troops|
The level dependent attributes (such as construction costs and generated culture points) can be found here.
For the full table of construction times click this link for server with 1x speed and this link for server with 3x speed.