Dominatrix pretty much hit the nail on the head. For me, the switch to Republic always seems to result in my economy bottoming out, at least in the short term. But once I get my empire set up for Republic, there's no need to switch to anything else. By the time I get all my specialist farms up and running,
Unit Support: Which version are you playing? In vanilla, Republic offered no unit support (IIUC), but in Conquests, Republic supports 1 military unit per town, 3 per city and 4 per metropolis. So getting a few of your towns to hit size 7 or above goes a long way towards unit support. Add some markets, both for the $$ and the happiness. The commerce bonus (+1 commerce to every worked tile already producing at least one) goes a long way to supporting more units, especially if your citizens work lots of roaded and riverside tiles.
The other common complaint about Republic is War Weariness, but it's not so bad if you manage your losses in war. Republic is a little more fiddly than Monarchy, which is generally considered good for the All-War variant, where you can never make peace.
Feudalism: I have to admit that I've played relatively little with Feudalism, so I don't really know much about it. But here's what I know. Feudalism is generally considered good for culture wins, where you can pop-rush the culture buildings. Its "inverted" unit support plays well into this, where towns actually provide more unit support than metros. Need a temple? Grab the whip. Paying too much in unit support? Grab the whip. In the few games (maybe only one, even) in which I used Feudalism, I have to admit that I was surprised by the military that my meager empire could field.
Communism, Democracy, Fascism: Never played any of them, but here's what I've read. Once you've settled into Republic, Monarchy or Feudalism, unless you're religious, there's no sense in switching. Communism comes too late in the game and has to be set up entirely differently from Republic. Democracy comes late in the game and is an optional tech, preceded by other optional techs (lots of unnecessary research to get to it). Fascism. All I remember is that I haven't read anything good about it, but I'll leave the details to others so that I don't give you any bad information.
Run a search through the fora and you'll find lots and lots of old threads on governments.