Training challenges:
Beat Tournament mode to unlock all training challenges.

Trainer team:
Beat all Academy challenges to unlock the Trainer team.

Egyptian team:
Beat the Alexandria level on Advanced difficulty in Tournament mode to unlock the Egyptian team.

German team:
Beat the Germania level on Advanced difficulty in Tournament mode to unlock the German team.

Gladiator team:
Beat the Circus Maximus level on Advanced difficulty in Tournament mode to unlock the Gladiator team.

Thanks to Revolution reader Ryan Max!