Before you start, create a system-level regression test suite for the application. You need this so that you can verify that your changes don't break things.
To do the refactoring, you want a use a combination of a good IDE, and text search tool (e.g. grep). Use the text search tool to find occurrences of the "syndromes" that you want to fix, then use the IDE (and its builtin refactoring capabilities) to fix the instances ... one at a time.
For example, Eclipse allows you to rename a method or class, or generate getters and setters. So you'd cure a 'public' attribute by:
- Change the attribute to private.
- Generate the getter and setter methods.
- Save the file.
- Go through all of the Java compilation errors resulting from the fact that the attribute is now private, and change to use the getter or setter as appropriate.
This approach will give you the low-hanging fruit. More fundamental design issues are more difficult, and may be impossible to fix without fundamental restructuring of the application. The refactoring capabilities will help you execute such changes, but deciding what to do is ultimately up to you.
Finally, my advice is to not be too ambitious. Go for incremental improvement, and be prepared to draw the line when the code is "good enough". You won't achieve perfection ... not even if you start from a clean slate ... so don't set your expectations high.