Most will go for Monarchy first becuase it's easy to get to early (and get out of despotism ASAP) and because it won't penalize the player for having up to 3 units or having those units outside of their cities. The support for a settler is also (I believe) a single food only.
This allows a player to fight early wars without happiness issues and, for me, settling all over the place without the settlers becoming a huge drain on my production.
The switch from Monarchy to Republic (or straight to Democracy for some) is always a touchy issue. Two things that work against you in Republic is happiness and production. As duke mentioned, ships that are normally used for transport soley (trireme, caravel) will cause some unhappiness or restrict their supporting cities from sending any additional units anywhere - not to mention any groud unit outside of its city. Those units will now always require shield support as well, and if your city is producing 4 shields with two phalanx and a trieme to support, all of a sudden it is a mere 1 shield.
Of course, the benefits are wonderful in a Republic/Democracy so it's worth switching. Celebrating your cities so they grow, excellent trade dollars and much speedier research make these governments worth every effort.
Pretty much all depends on the level you play and the situation of the world at the time. If in a war, I'll sometimes stay in Monarchy for a while longer than usual. If on Deity, the same can hold true as I feel the need to get Mike's Chapel and another happiness wonder (at least) before the new government pays off.
Only a few examples of the myriad that could happen. It is really a pure judgement call that you'll feel more comfortable about as you play more and more.