Major Tech Push

[DreamSlave2112]

Aside from building more cities and trading techs whenever possible, how else can I get maximum techs? The major sticking point for me is which tech lines to pursue first. In Ancient times, should I get lit first for libraries or Republic for the extra commerce?
Likewise in the middle ages, should I go for democracy first or theory of gravity (for Newton's uni)?

[Verrucosus]

For the ancient period, Literature should come first. This is applies even when you don't have a lot of contacts and the Great Library is not as valuable, because (in my experience) the Republic advance takes a long time to research and constructing the libraries in the meantime helps both to speed it up and to have the library multiplier already in place when the additional trade comes in.

For the middle ages, I'd say it depends. Democracy is important in fighting corruption and very helpful for larger empires. On the other hand, dropping Newton's College into your science city (particularly on top of the Colossus and Copernicus' Observatory) isn't exactly marginal either.

(Also, unlike Democracy and its prerequisites, Theory of Gravity is a requirement for advancing into the industrial age. If your research is good enough to begin with, the benefits from Democracy may not make up for the time lost in the detour. If starting industrialization as soon as possible is your priority, it's even an option to go back to pick up Democracy only after you've started to build railroads and factories. That's no longer the pure research perspective, though.)

Verrucosus

[Kloreep]

Maximizing your research really depends on your situation. You do need optimal empire size (a palace core of cities + more around a Forbidden Palace) to get all the commerce you can. If you find yourself cramped and needing to fight wars to achieve this, you'll have to build troops to start with instead of libraries, sacrificing short-term as a long-term investment in a good-sized empire. In such a situation, you may find yourself fighting Despotic wars, while research Republic to switch to after you eliminate a civ or two and are able to lessen your army size relative to your number of cities.

That said, if you're able to expand relatively peacefully, you may be able to have the luxury of building libraries early; in that case you'll need Literature. That said, early libraries aren't always a great investment. You do need your cities at decent sizes to pull in enough commerce to make libraries worthwhile - you don't even profit until you get 4 beakers reliably, and then it's just a +1 beaker gain - and if you don't have enough workers to keep all the tiles you're working roaded and irrigated/mined, workers are probably a better investment than anything a library could get you.

It also depends somewhat on traits. If you're Scientific, libraries are a lot easier to make an early investment in, especially if you have good enough happiness to go without temples for a little while. Literature becomes a lot more attractive.