Throughout the course of the game, there are a few different civic strategies for a civilization to quickly raise up a military.
The earliest civic to rush out a military is Despotism. Pair it with Conquest when available, Despotism peaks in the early eras, because military units, with few exceptions, are almost all under 60
. Meaning you can whip them out with 2 population or less, which is fairly easy for most cities.
The second civic strategy comes with Tributaries. I generally consider Tributaries to be superior to Despotism because 1.) It frees up a government slot, and 2.) It doesn't accrue
, which, depending on your civilization, can quickly become a problem.
The third civic for rushing out a military is Citizenship, and buying mercenaries. For most civilizations, this strategy becomes most attractive in the Renaissance era, where international trade routes and trading companies can bring in much more
than you had in the previous eras. For a couple of civilizations (Ex: Vikings with their UP, Arabs controlling several holy city shrines), Citizenship can be attractive in earlier eras as well, as they have a plethora of
to play around with.
The final civic for rushing out a military is Nationhood, the good ol' draft.
To answer the other half of your question though, I would run Monarchy/Colonialism once my empire is already established, and my core (after keeping it small with Tributaries) needs to start growing to support peripheral population.
Theocracy is also a very attractive civic when quickly raising an army... The only problem is it's in a very awkward position on the tech tree. Usually I rush through the Medieval era to Firearms, and Doctrine is usually forgotten until I'm done growing my military. If you can swing it, however, Theocracy can pair nice with Tributaries. After all, you aren't constructing buildings with Clergy, and you aren't supporting any specialists with Monasticism.