In terms of the general quality of administration, republics are a superior form of government to monarchy. This is why republics can easily degenerate into monarchies by trashing their RT, but monarchies cannot easily elevate themselves to republics, even if their legitimacy is zero (people in this situation don't believe in your king, but they still want some kind of king). You can also think of Republics as a high-energy state and monarchies as a low-energy state - decaying from republic to monarchy is a bad thing in the long run, but in the short run a lot of 'energy' is released in the form of bonus monarch points, and as a result you can accomplish some spectacular things on the way down.
The main weakness of Republics are that they can end up with high stability costs due to RT, they don't have the diprep bonus that comes with high legitimacy, they can't become Holy Roman Emperor and they can't play the PU game. But if you don't care about HRE/PUs and you take the right ideas, none of these is a major problem.
Your RT will usually drift along at a medium level, because you want to re-elect reasonably often, but also you want the events that give +25RT (I think they only trigger when your RT is somewhat low). It's not hard to manage your RT, but you do have to be sensible about it, especially with the frequent events that will encourage you to sacrifice your precious liberty in return for some one-off comforts. Dipping into dictatorship is risky (you could degenerate into a monarchy) but potentially quite profitable if you have a youngish ruler who has been re-elected a few times. Anything that gives you more RT is very valuable, as it will allow you to re-elect more frequently.
Noble Republic is pretty weak as republics go. You're better off with a government that has a short election cycle, because shorter cycles mean lower re-election costs and more chances to upgrade your president before he dies of old age. Oligarchic is better, and then you may or may not want to switch to Administrative. In the late game, you'll definitely want to switch to Constitutional, unless you somehow manage to become a Revolutionary republic.
