Ask lots of questions- not just of the stage manager, but of other folks on the crew. Making an effort to learn names, and chatting with each person for a minute can got a long with towards having a good rapport with the ASM and crew. The corollary to asking lots of questions is: Take lots of notes! In my experience, I usually get 1 or 2 days of shadowing the SM, then 1 or 2 days of calling the show with the current SM shadowing me and giving notes. Then I'm on my own. The more notes I have the better off I am when I'm on my own.
I usually try to keep things consistent for the actors and the crew by recreating the previous SM's way of doing things, particularly if I'm just subbing in for just a day or a week. If they are used to getting the places call at X minutes before curtain, I'll do it the same way. If I'm taking over after opening and calling the entire run, I'm more likely to do things my own way.
If there's a tricky sequence to call, see if there's an opportunity to do a dry run before doing it in front of the audience. I've lucked out before and was able to call a put-in rehearsal in the afternoon, before calling the show in the evening. Sometimes it's possible to run a sequence with just the crew right before the house opens. I also like to come in early if I can, and practice calling anything tricky on my own, especially if it's a sequence with cue lights that involves hand-eye coordination.
What Ruth said about knowing the show is really important. Understanding the "why" will make you much better prepared to run the show on your own. It also will let you better project confidence- the cast and crew has to trust you and to know the show is safe in your hands. I always ask the current SM if there's anything they want me to look out for/keep an eye on. (This actor usually runs late, curtain has to go up by 5 after or the orchestra goes into overtime, a certain prop preset has to be triple-checked, etc).