OK, first reply reply to your post.
I am usually REALLY certain nobody will beat me only when there is one turn left to its completion (bad experiences )
Usually, playing on Monarch, I go for Republic at a steady pace (15 turns), since it seems it's one of last techs researched by the AI and I want to get out of Despotism asap.
In the meantime, I sell techs around to get plenty of cash. Once into the Middle Ages, I usually have 700-1'200 gold, then I beeline first for Gunpowder, via Engineering, then switch for Adam Smith (which for me is the more important Wonder in the Middle Ages).
When playing on archipelago, I often leave a group of civs technically backwards, to milk them at leisure, specially if I plan to grab soome land from them.
Oh, I forgot to mention that my second town goes for the GL immediately, after building a worker.
All the rest is subject to the whims of the game.