Most of what needs to be said has, beginning with duke's reply. Assuming SP Deity, I'll just add that early on, it really pays to have in your mind what you'll be planning, and then prepare for Republic if that's where you're heading. The most dominant factors are terrain type and enemy pressure.
1. Terrain type. If you're on an isolated island with decent terrain (e.g., lots of rivers, grass, plains... not dense mtns, ranges of hills, etc.), then you're primed for early Republic... you'll likely be needing the science anyway. On a continent, if you can choke out or block AI expansion, republic is an option.
2. If you get Maytagged (you get in long battles, over and over and over, and get knocked around and around), stick with Monarchy, build units, expand with those one-food settlers, and try for a direct democracy later on.
Either way, the decision of early republic or continued monarchy (ER or CM) should reflect your need of units and terrain in the early game. The option of direct Republic is one that I personally don't use (I always get Monarchy ASAP).