In order to progress down each of the different tech trees, you will need to research both vehicles and the different module the vehicles can mount.
These modules feature better guns, engines or turrets.
You can research new vehicles in the Tech Tree, accessible from your garage (with RB).
You will want to start by selecting the Nation of the vehicle you would like to unlock.
Next, make sure that you have researched and upgraded all the tanks leading up to the vehicle that you wish to research by clicking on them and confirming that you wish to research them.
It is important to note that it does not cost silver to research a vehicle, or research modules. The currency used for this operation is experience specific to that vehicle, combined with «Free XP».
You can upgrade a vehicle from the garage by first selecting it, and then by pressing A to access its menu.
Select «Upgrade» and purchase all of the upgrade packages that lead to the tank that you wish to unlock next. It is necessary to perform this step before you can research the next vehicles in that line!
Once you have researched the upgrades to the tank that comes before the vehicle you want, you may go to the Tech Tree to research it. In the following image, all of the tanks in green have been researched; the tanks in grey have been researched and purchased.
You're all set! Good luck on the Battlefield!