if you have fully researched lets say level one infantry you need to build an army base level one to produce that item.
1 you go to home cities and click one of said cities, then click the construction icon then click army base level one to produce level one infantry.
2 your going to need to also build a recruiting office for man power once you run out of man power you can make anymore troops.
3 you need arms industry to make more product from components to rare material , ( I STRESS THIS PART HERE YOU NEED TO CONTINUE BUILDING YOUR ARMS INDUSTRY UP, YOU CANT BUILD TO LEVEL ONE THEN EXPECT TO HAVE RESOURCE ).
4 you will have to do this with recruiting office as well
5 my recommendations build 3 army bases 4 recruiting offices and level one arms industry in every city and after your production has level built up more product. build recruiting offices level one in your in your remaining citys