I was having major problems with the Republic switch myself until just recently. There are a couple main causes of a failed Republic (unable to keep the population happy or unable to support army).
The first cause of a failed Republic is the overuse of pop-rushing during Despotism. Now I never pop rush in any city except those specifically designated for churning out units. These farming cities find the largest bonus food source available and set up camp. Pop rush that granary first, temple, barracks, and then units to your heart's content. Usually one of my first 4 cities will be a pop-rusher, planted beside a wheat or cow square. This city will never rise above size 2-3 so take that into consideration during placement. One big "trick" I've discovered is the liberal use of flood plain. I've realised just how amazing flood plains are and treat them so. PACK cities, two spaces apart, all along the floodplain. Make sure to place the cities in the outlying desert since you want to be able to irrigate the actual flood plain to its full potential - no need to be on the river since it will not be larger than size 2-3. Jamming cities along an irrigated flood plain in Despotism equal <b>massive</b> unit pop-rushing. It's glorious.
The second cause of a failed Republic is not enough luxuries and/or a poor infrastructure.. If you haven't conquered 1-2 neighbors and taken their luxuries before the switch you're going to have problems (in my experience). I generally try for at least 3 luxuries with marketplaces in all core cities before switching. Even then you may have to up luxuries to 10%, but if you've conquered enough land/luxuries that shouldn't be necessary.
In a related idea, I generall have three waves of war. The first comes just after the research of Iron Working, hooking up my cities to an Iron source, and upgrading all my Warriors to Swordsmen. I try for about 10 Swordsmen for my first invasion on a Standard sized map. The second wave comes from Horsemen, which I research directly after Iron Working. Hook up that Horse resource to your Capital and pop-rush the hell outta those things. You should have a few pop-rushing centers hooked up by this point, especially if you've started near or conquered an area with flood plains. The third wave comes after the research of Chivalry. Upgrade all those horsemen you've been rushing and it's all over for the 3rd and 4th civ.
I play on Regent difficulty at the moment, so keep that in mind, but I thought the above may help those newbies out there. Actually, using the above strats, especially milking Flood Plains, Regent has been too easy. Monarch, here I come!
Peace