I've never switched to Communism or Fascism, so I can't really speak to them. However, I use Republic pretty consistently, and it's much more "fiddly" than Monarchy. In particular (or at least for me), Republic always suffers some "growing pains," when I first make the switch. I usually make the switch pretty early, so my support costs jump, and I have to move the lux slider up, but it gets me out from under the despo penalty. It's a short-term cost for long-term benefit, IMO.
I agree with Nergal when he says:
The key to it is commerce.
As a cash-rush government with no MPs, Republic boils many major game issues down to gold. Need that barracks right now? Cash-rush it. Need saltpeter? Buy it. Is it just "way too crowded?" Increase lux spending.
The good news is that Republic also provides the commerce bonus. The trick is making good use of the commerce bonus. Because of the way the commerce bonus operates (+1 gold to every tile already producing one), it's important that, well, you've heard it before: roads, roads & roads. By my thinking, you'd want roads pretty much everywhere, for the +1 commerce, movement and for resources that may surface later. Under Republic, though, the commerce bonus adds an additional gold to that road.
In the hinterlands, corruption will eat up all that additional gold, but in the core, all that additional uncorrupted gold gets channeled through markets, libraries, etc., and really begins to make a difference.
In his post above, Nergal makes some very good points with regards to Republic. He also makes a few statements that I don't agree with.
There is no MP limit so garrisons are a waste of time. Maybe 1 of the strongest defensive troops you can build in every town, at maximum perhaps 2. Unit support costs are a killer. If you have a border with another Civ then perhaps stock more defenders there and leave some of the core towns empty. As I said garrisons have no effect on happiness.
This is true. The only thing I would add to it is that when I make the switch to Republic, I usually don't have much that
isn't a border town (or frontier town, at the very least), so almost every town still needs at least one defender, at least on my initial jump to Republic. It's not until my civ gets a little better developed that I'm actually able to disband those defenders (or at least keep them moving with the borders).
Juggle your entertainment slider. Check the cities and if they arent producing commerce and beakers then take citizens off and tuirn them into specialists. As long as the town is still growing that is, after a while it reaches a balance where it can still grow and support tax collectors and scientists. Push entertainment up until your productive towns are not going to disorder on you. Then raise research. If all is going well then you should be getting more actual beakers per turn in research than you would under Monarchy, even though your research % is lower. Thats the bit I had trouble with as nobody really says that bit.
I can't quite tell what you're saying here, Nergal. Are you talking about using specialists in the core or out in the hinterlands? If you mean the core, I disagree. A citizen working a tile in a core town with a library will often produce more beakers than a scientist. If you mean out in the specialist farms, that's a different story. In that case, I agree. Specialist farms are a very powerful tool.
Build marketplaces, it ups the commerce output and provides extra happy faces with luxuries. Temples and Cathedrals are good, possibly Colisseums, and any Wonder that provides happiness. But markets are a priority. Happiness is the biggest downside of Republic, its a total pain not being able to MP.
I agree with Nergal on markets, but have to disagree on temples, cathedrals, coloseums and (possibly) Wonders. There's a time and place for a temple, and the happiness buildings are, obviously, fine for culture wins, but if all you're interested in is stomping the snot out of your neighbors, they're not necessary and quite possibly detrimental. One trick in dealing with Republic is keeping your empire lean. Don't build things you don't need. Once you build the temple, you pay upkeep, whether you really need it or not (barring selling it). On the other hand, if you trade for (or capture) a couple of extra luxes, you can probably lower the lux slider. As for Wonders, honestly, I almost never build the happiness wonders, so I can't say that they're most definitely a bad choice, but I seem to get sufficient happiness using cannons and cavalry.
War, I hear you cry, what about that. Be very careful, dont get any troops killed and dont lose any cities to the enemy. Wage short term wars. . . . . With long term wars the WW gets out of hand.
While I wouldn't go so far as to say you can't afford to lose
any troops, you do have to be much more careful about it than with a non-WW government.
Long-term & short-term, I suppose, are relative terms. As long as you avoid heavy losses, you can war in Republic for virtually as long as you like, especially if you:
. . . . war like a chicken . . . .

I'd never thought of it that way, but I can't say that it's totally devoid of some truth! Part of waging war in Republic is managing losses. Sometimes that means not attacking, or making peace until you can get your forces organized.
Anyway, Republic takes some fiddling. No doubt about it. Once you get used to it, though, it's not that bad. Decent unit support (once you've developed somewhat), durable enough for most warmongering, and lots of gold.
Also remember that the (C3C) support structure for Republic is 1/3/4. That means that once a town (up to size 6) hits city status (size 7-13), its support triples. So getting towns to hop that barrier is important.
Edit: X-posted several times over.