Class and object design can be a tricky thing. It’s not really something that can be taught or quantified. This skill is developed through years of experience. This may be more of an art than it is a science. No expert can come up with an exact formula on how this should be done but all experts can pretty much agree on what is a bad design.
A possible starting point of a process could be to develop a written problem statement outlining the issue at hand. This can be done by interviewing a customer or whoever the requirements are coming from. Once the problem statement is developed you will find that the nouns in the statement will turn into classes and the verbs in the statement will turn into operations. From there relationships between the classes can be derived. Statements like “is a” can denote inheritance, “has a” can denote composition, “uses” can denote delegation, etc.
A UML diagram should be drawn modeling what was discovered by the problem statement. It is also good to keep a catalogue of design patterns at hand so that a given pattern may be applied to a module of the software system. Remember that Object Oriented Software is about loose coupling, encapsulation, and reuse, which design patterns help enforce. Once the initial diagram is developed it is really more of a guideline than concrete fact at this point. When the implementation process begins new facts will come to light that haven’t been considered before which will ultimately force the redesign some components of the software. Once this happens the diagram should be updated to reflect the new functionality. This will create a software cycle that will bounce back and forth from implementation to redesign approaching a final version of the software.
This is really only a generalized overview of an absolutely monstrous topic in which there are many different paradigms and many different ways to go about tackling the problem. As a closing thought, keep in mind that a design is never really complete, that it can always be improved upon and made more robust, clearer, and more efficient.