Software design is all about decontructing a system into manageable units, identifying relationships among them and getting the units to interact to fulfill system requirements. To become a good object oriented designer, you first have to replace the "procedural mindset" with an "object oriented" one. That, however is not as simple as it sounds. The best way to get there is by practice, lot's of it. Take every opportunity to create a good design.
When creating an entire system, think about the classes, their relationships, and interactions. When creating a class, think of what responsibilities it should fulfill. What should the API be like? How does it relate to other classes? When creating methods, think about the method signature and about the code in the method. Does the method do one and only one thing? What exceptions should it throw? Is is intuitive and easy to use? At whatever level you are creating the system, you can always think of design. Remember, there is a lot of power in the process of understanding, reflecting, practicing, and writing. Use it to learn continuously.
The natural progression after understanding basic design principles is understanding design patterns and then enterprise architecture. Start with the "Gang Of Four's" design patterns and then move towards more complex topics.
A few good resources for for reference: