My experience was a little different from the norm. I shadowed an attending at a university hospital, and he basically had me shadow his residents while he did "attending stuff".
I liked it because I got to shadow a different person every day, so I was able to pick the brains of multiple anesthesiologists. Whoever you shadow, you should try to pick the brains of anyone who will talk to you. And maybe you can sneak in with some of their colleagues and observe a couple extra inductions/recoveries (the part of surgery where the anesthesiologist does the most).
Also, as for shadowing anesthesia in general... I saw a wide range of procedures from multiple medical disciplines. If you were shadowing a single cardiologist/gastroenterologist/orthopod/etc, the variety you would see would be smaller, but you would be able to learn things in greater depth.