This kind of question is better suited for the
Quick Answers thread, but I'll answer it here.
First of, realize that science is driven by commerce, and commerce is driven by the amount of tiles
and how your work them. It does no good to have 20 people out there working tiles if you haven't improved them. So make sure all of your tiles are at least roaded. This will increase the commerce you make on each tile, increasing the amount of money you make and the amount of money you can spend on science. The more money you make, the more improvements you can build, the larger army you can support, and the more you can devote to science.
Basically, you need to make sure that you have at least marketplaces in all cities. Not only does this increase the base commerce by 50%, but it also increases happiness. At 3 to 4 lux's, you get two happy faces each, 5 to 6 three happy faces each, and 7 to 8 four happy faces each. This will make your city happier and you spend less money on the lux slider. Banks will also increase the base commerce by 50%, as well as Commercial docks (in PTW.)
Also, your government dictates how much commerce you make. Well, not reall, but it does influence it. In despotism and anarchy, you lose one commerce per tile worked. In communism and monarchy there are no penalties or bonus', and in Republic and democracy you get one extra commerce per tile. This can add up quite a bit.
So basically make sure that you have marketplaces, your tiles are roaded, and that you get out of despotism. This should be a good starting point for you, and you can expand on these concepts more as you learn the basics.
Hope this helps, and good luck. Feel free to ask again if you need more clarification.