Here are some suggestions that might help. Specialize a couple of cities to only do science. These should have lots of water tiles to work. In these science cities, make sure they have all improvements that increase science, like the library or university, and then settle any great scientists there. If you can build a wonder like East India Company, it will really help (+1 trade per sea square). When researching techs, go for the ones that give you a science bonus if you're the first to get it: Literacy (+1 Science in every city), University (+1 science in every city), Atomic Theory (+2 Science in every city), Electronics (+2 trade in every city). Make sure you're in Democracy to get the extra trade bonus.
Once the population gets higher in these cities, they should be producing around 500 science per turn each. Over 1000 science per turn for each city is possible if done right.
Another tip: keep yourself at a contant state of war if possible with the other civs the whole game. As long as you're in the tech lead, you should always have defensive units advanced enough to repel any attack. With them at war with you, they will focus on sending army after army to be slaughtered by your defensive units, and they will all fall way behind you in tech.
Make sure that you have developed 2-3 cities specialized for production (near hills, mountains), with all the improvements needed for production (factory, iron mine, workshop, etc.). When you have the necessary tech for space travel, switch to communism (for the production bonus) and build/launch your spaceship fast.
You can also have some cities specialized in Gold instead of (or in addition to) production cities. Then you can just buy each space ship component each turn and quickly launch your ship.
The key is getting your science cities going early enough in the game so that you're not getting close to the game ending on turns.
Hope this helps, good luck.