Good advice so far, but also be receptive to what your resident expects from you. Offer to do whatever you can, but don't get in the way, and don't be too persistent.
I tend to be very particular about certain things, and I don't want someone else in my business. I'm happy to talk to you about almost anything, and let you handle the airway, etc. if I think you are capable. Ask any question you want, and I'll do my best to answer.
Unless you'll be with me for a few days, I don't want you to (mess up my) chart, and frankly I probably don't want you to touch my art line. And I would never ask you to empty the foley bag, 'cause that's my job, and it's not very educational for you. I will rarely let you draw up any drugs, but I'll probably let you push.
Point is, we each have different personalities. Read the personality of the person you are assigned to, and act accordingly. If you can arrange to work with one person for consecutive days, try to do so, because you will get to do more as they recognize your skill. But recognize that some of us like our own little personal space at the head of the bed, so if someone suggests you go take a break, just take the break.