On the Governer page, you can increase your overall spending, and increase spending in a particular area of manufacturing, both of these will increase your manufacturing. For example, many people increase spending to 100% (it takes population and improved governement to increase this max level) right at the start, as well as shifting most if not all spending to manufacturing. That way, your next colony ship can get done in 3 (says 2) turns.
On Starbases, you can build modules which give manufacturing bonuses, they are listed in the second and the bottom button when a new constructor arrives. More starbases to help, but there's a limit beyond which extra production gets harder to come by. When you feel ready to worry about that, I suggest reading about "Economics in GalCiv" in the Encyclopedia first.
A little bit to start on