What's the quickest way to build up troop numbers in my armies?

To increase the number or troops you can place in an APC army, you need to level up the barracks building, unlock the 3 heroes slots, place high level (preferably orange) heroes and unlock and upgrade their troop number skills (the first one, Dictator, and the 6th one Awaken)

The fastest for you will depend on where you are currently. But as all of them require levelling and acquiring resources or medals you just need to grind (or pay) and try to work on all of them. If you already have the research done for 3 hero slots then make sure you place 3 orange heroes and use them to increase their level. Do what you can to gain wisdom medals and work on their skills.