How do I research class skills?

All research is done in the Institute building and you can also get to the research options via the small flash icon on the upper right of the screen when you are in your city. Every research that you do gives research power / points. You can see how many points each research item will give by tapping the small info icon to the top right of each research item screen.

I am not sure what you mean by class skills though. If you mean the skills of the heroes then these are unlocked and upgraded in the heroes menu. You need wisdom medals to upgrade skills and medals money and also sometimes hero medals to unlock them.

If this doesn't help please explain your question in more detail.