Ask lots of questions and just make an effort to learn. Your dentist will probably want to share some of his/her knowledge with you. Definitely make sure not to be a pest with your questions by asking too many!
When you shadow, you just kind of stand there and watch them interact with patients and watch them do stuff in their lab in between patients. Sometimes, they will be explaining to you what they are doing as they work on patients to help you understand what is happening. Over time, after having watched a huge number of procedures, I started to understand the routine for most of the common dental procedures. I think that the amount of things that they let you do is a reflection of how much they trust you and like you...just definitely make an effort to be polite and nice (since time flies by so much faster when you have something to do). Even something like filling out the chart when my dentist was doing pocket depth probing was of fun.
In regards to asking you to assist patients, usually that's not necessary since there are already assistants to usually do all the suctioning, etc.. You can of course help here and there, though.