The little beaker icon in the top left corner of the window takes you to the Tech tree, where you can select which technology you want to research. Once you've selected a tech, you gather science points automatically. At the bottom of the tech tree window is your progress bar, showing how many points you've gathered and how many more hours/days you have to go until completing your research.
Science points are gathered in your shrine of knowledge - clicking on your shrine shows the percentage of your resource income that is put towards science, and how many science points you're currently collecting per hour. Thus, you can increase your science income by upgrading your resource production (crops fields etc) and by upgrading your shrine of knowledge.
