You can't research more than one thing at a time.
There's several ways to increase the speed that you research techs.
1) Increase your overall funding level. Really, you should try to balance your play so that your spending is at 100% at all times, with rare emergency-type exceptions.
2) Increase the amount of that overall funding that goes to Research.
3) Build Basic Labs/Xeno Labs/Research Centers/Research Academies/Invention Matrixes/Discovery Spheres on your worlds to increase the overall amount that you can spend on research.
4) Take a Research racial bonus or pick the Technologist party as your political party.
5) Find and develop Research Resources floating around in the galaxy.
6) Expand your empire. Having more worlds means more research can happen.
As for modules, I assume you mean starbase modules. By moving a constructor onto an established starbase, you can add modules to the starbase. These modules fall into many categories. Any starbase can take modules that help it in combat (offensive or defensive value bonuses), and depending on the type of starbase, it can also take resource mining, influence magnification, military, or economic modules.