The best way to advance your technology, extract metal, and terraform planets is to keep your spending nearly constant over a period of time. Make your adjustments gradually. A small allocation that continues over a long period has a much greater effect than the same money spent in a few turns. Range tech should get some money, too, and if you have extra cash spend it on radical tech. Be sure to spend plenty on weapons and shields, even if you don't expect any battles soon.
Mini tech is less important in the beginning, but it becomes more important as the game goes on.
Speed is very important because the faster fleet shoots first in a battle, and that can be the deciding factor. In circle, random, and spiral galaxies, the central portion is the most desirable and hence the most bitterly fought region of the galaxy. If you colonize any planets in the central region you must defend them strongly. This is true to a lesser extent in the grid galaxy. The ring and cluster galaxies have no central region, so you can only be threatened by one or two enemies at a time.