Plan a trip to the park during the off-season. Early spring and fall are the park's least busy times, making it easier to find characters and enjoy their company (See Reference 1).
Mark places on the park map where you are likeliest to run into characters. For example, mark Toontown Hall of Fame, where you can generally rely on finding a variety of characters. This will help you plan your various stops throughout the day (See Reference 2).
Visit City Hall in Town Square and speak to some park employees. They have access to information regarding the whereabouts of different characters, and can point you in the right direction (See Reference 2).
Approach a character with your autograph book. If you're shy, asking for an autograph is just one way to break the ice---you could also ask your favorite character to pose for a picture with you (See Reference 1).