In the training menu you choose which employees to train in which course, in how many rooms (1 room has space for 10 employees. Which vehicles need which training is written in the vehicles vehicle market entry (where you buy vehicles).
You can choose to let alliance players join your courses by setting a delay on the starting time. You will earn the amount of credits per day per employee training you specified for every one of your alliance members employees. The training is free for your own employees. Lastly you choose which employees to send. Remember that the employees training aren't available for duty! Using the button in the bottom right corner you can see how many free places there are left.
When having started the course you can view and manage them from the existing lessons tab.
Expanding an academy allows you to train more personnel at the same time.