It really depends on the military unit as to how they will behave on the terrain. Progressive Era can be quite a challenge not only because of the road/city changes, but also because the way the military units behave changes. For example, conscripts (light unit) now get a bonus against tanks (heavy unit), whereas in ages before heavy units get a bonus against light units. Conscripts also get attack/defense boosts in the trenches.
The different aspects of the landscape will affect movement costs, which is how far your unit can move during its turn. I personally do not know the exact effects of each off the top of my head, but I don't feel it's necessary to win these battles. As in prior ages with battles my strategy is to come up with the best offense to attack the defense. Know which types of landscape will give your unit an attack or defense bonus and use those to your advantage in the battle.