Congrats on the new gig. If they have extended an offer to you, then you've already proven your worth, so don't worry that you somehow might not be pulling your weight. You're in.
As for planning for a successful and productive career, I would offer only two suggestions to consider:
1. Work hard. There simply is no substitute, and speaking as manager myself, I want to be able to count on my staff, and know that if I give them a project, it will just get done. Period. Be that guy/girl.
2. Cover the holes. What I mean here is that in every organization I have ever seen, there are certain areas where things just don't work right, or end up being a lot more difficult than they need to be, or simply would benefit from changing the workflow or business process. If you find one of these issues, and you probably won't have to look very far, consider them not so much as problems, but as a chance to get an excellent performance review and fat salary increase. If you can devise a better more efficient process or automate something that ends up simplifying and/or improving the jobs of your co-workers, particularly the group you work most closely with, you *will* get noticed in the most positive way, and don't be surprised if you find yourself getting invited to meetings with higher level people to ask for your opinions about larger, more important projects. Rinse, lather, repeat.
Either way, congrats on the new gig