The best way to mass units early in the game is to research Bronze Working and chop forests. The forests will provide the closest cities with hammers (production) and this will speed up the manufacturing of troops.
Building mines and farms (don't forget work boats for coastal food as well), especially on bonus resources, will also help you reach an equilibrium of city happiness VS health (not so important) VS hammers VS food. If you have excess food but people are getting unhappy, you can choose the Slavery civic and start whipping the people for hammers. This will reduce your population to help control happiness (if done right) and will once again provide hammers for manufacturing. However, I always like to think long term and use this as a last resort. If you have Monarchy, you can adopt Hereditary Rule and should be able to control happiness if you're also manufacturing troops at the same time.
The strategy will be slightly different in each game (i.e. sometimes I leave forests to wait for the 50% chopping bonus with Mathematics and then use hammers for building specific Wonders) and the circumstances. Civilization is also played very differently depending on difficulty level, map sizes, map types, game duration, humans involved (multiplayer), randomness (i.e. unique opportunities that a game may present), AI weirdness...etc...
That's why Civilization is so great! Enjoy the challenge of making the most of what you have.
