At the start, I use a training family (champions/hunters/riders) to produce units. Their cities start with 10 training instead of 8, which means 6 turns to produce a warrior/slinger (60 training) instead of 8 turns. Champions family seat gets an extra +25% training, which means it starts at 12.5 training (15 with Rome). Once I research military drill I aim to have workers ready to build two barracks in the training city. They each give +2 training so together they boost the city from 10 to 14 training, which means about 4 turns per warrior/slinger.