Don't expect all your cities to be producing expensive units like Modern Armor very quickly. Instead, specialize your cities and have them do what they do best. I'd be willing to bet that the city that takes 90 turns to build a Modern Armor has very few (if any) mines, windmills, or production resources, and if it does, it probably doesn't have one or more of Forge/Factory/Power.
Specializing cities is more of an art than a science. But in time, you'll learn to recognize that (for example) a city with lots of hills, a production resource like Iron or Coal, and a few good food sources (especially Fish) is an ideal production city. Improve and work those food specials, growing the city as fast as you can. Improve the production resources, and build mines until you can't grow any more.
If there's no obvious site rich in hammers, you can improvise later in the game. A site with lots of grassland/plains can be quite productive with railroaded lumbermills and (post Biology) irrigated grassland farms (although a food resource will really help). And as a worst case, you can go with lots of workshops, although I prefer to keep my forests and build lumbermills so I don't get the -1 food.
Anyway, once you have 2 or more of these cities rich in base hammers up and running, put Forges, Factories, and power in each one. Ironworks for one city; Heroic Epic and West Point in another. Barracks and Drydocks for all. These cities will produce modern units very fast - perhaps as fast as 1/turn. They'll do the bulk of your unit construction. Any city that takes longer than 15 or so turns to build a unit would probably be better off building something else.