Design, unlike coding itself, is a heuristic process. There is no "perfect" way. You have to know what, how, when to use and yet you will still be wrong sometimes. It is a wicked problem. You cannot find a solution till you've solved it. You return to revising your design several times in a project(hopefully, fewer and fewer times as your experience increases). There is no advice in design to be taken for granted. Everything should be weighted against your project and you should always remember that your primary goal is to Manage Complexity(for more about this, read Code Complete 2nd edition, it is the source for many of my posts). These being said, here are some guidelines for problem solving.
1) Seeing the problem form multiple points of view
Very often I see programmers writing lots of heavy code just because they overlooked a much simpler solution. Never stop at your first idea. Probably the next one will be better. Learn to think from multiple point of view. Be creative. Design is heuristic, so it benefits from creativity.
2) Divide and Conquer
Learn to break the problem in smaller subsets, then think about each one separately. This way you reduce complexity
3) Top down
Start with a very abstract model of the system, then work your way down. It is easier to understand the puzzle when you have the big picture. Yet there are times when the big picture is not obvious...
4) Bottom up
Start with concrete cases, then try to work your way to the big picture.
Very often, you need to test your theory. For example, if the client asks for a database than supports 1000 transactions per second, be sure you create a quick and dirty model that tests just that on the database engine of your choice. Lots of design errors come from untested theories. Think of your prototype code as throwaway code. This helps you ensuring that you do not go into unnecessary detail. Prototype only what it is needed.
6) Team work
Often, someone else can see something you are overlooking. Be sure to ask for opinions as often as you can, it works wonders.
It is clear that the techniques I described should be used in a mixed way.