Playing a huge map won't help you tech any faster, as tech costs scale with map size. In fact, it will slow your tech down if anything, as your rate of expansion does not scale with map size, so it will take you longer to expand to any given % of the land area.
It is important to keep an ey on the situation around you and be flexible, but certainly there are some tech paths which are more conducive to fast research than simply casting around randomly.
In the ancient age, the primary path in C3C is Writing > Laws > Philiosophy > Republic (free). After that, you need concentrate only on Maths > Currency, and you should be able to trade for the rest. In PtW, getting into Republic requires a bit more effort. There are a couple of other stray techs which you may need to stop out for; Pots, if you can't trade for it; Mapping, if you are isolated; Literature, as soon as your core towns are in a position to benefit from libraries (i.e. large and well-improved).
In the Medieval, concentrate on the upper branch at least as far as Astronomy, remembering to build universities in your core, and prebuilding Copernicus. How much of the bottom branch you need to research depends on how your AI is getting on, but you should concentrate on getting to Gravity as fast possible, again prebuilding for Newton.
In the Industrial, you will probably have to do most of the work yourself, so the order is not so important. Prebuild Darwin, arranged to finish the same turn as a tech, and take Atomic and Electronics free. You need to make a judgement call on whether your empire needs hospitals to hit 4-turn research; Sanitation is an optional tech, so don't research it if you won't need it. You should be able to arrange for the AI to research Replaceables for you, but again there's a judgement call whether you want them to have such a powerful tech, and whether you can afford the delay of waiting for them to research it (often you are still railroading, and it will increase your worker speed, or you are still conquering, and artillery crush all opponents).