There are many electrode materials available, however, how do we decide which kind of materials can be used? Is there a way to look for the right materials using the electrochemical series?
It depends on analyzed compounds and range of potential.
Platinum of glassy carbon are quite good for working electrodes. Argent chloride electrode is quite common as reference. Good results can be obtained with gold electrodes however thiolic compounds can passivate gold surface and alter signal.
Agree with Elza's comments. When you talk about electrolysis, you need to specify what electrolysis you are thinking of. The choice of the electrode(s) depends on the ions in solution, the potential window and what you want to get out of it. The choice of the electrodes thus depends on various considerations.
Although most comments are right and valuable there is a difficult question to answer properly here. Many readings and some expertise are required to choose the right electrodes for a given reaction/analyte/solvent. If you specify these last items, may be we could help in your problem...