1st Scrum Master role....
So I was just offered my first actual Scrum Master role. I've played the part earlier. But this brings a whole new level. How do I know I'm doing a good job, and being effective for the team?
The Scrum Guide outlines the services a Scrum Master ought to provide to the Product Owner, the Development Team, and the Organization. Why not start with that?
Learn about, and practice, Level II and III listening.
Your answer is in the question you ask!
"How do I know I'm doing a good job, and being effective for the team?" Ask the team, get feedback from the team members. During sprint retrospective you can ask them how can i be of better service to you?
Talk to, share and communicate with other experienced scrum masters and implement the ideas in your role and reflect on the implemented idea get feedback and improve or try a new idea.
Try to meet other Scrum Masters, and ask for help and feedback.
If you have other Scrum Masters in your organization, they will be in a better position to challenge and assist you. But also try to attend meetups, conferences, or find places like this forum to meet other experts and get further insights.
Also regularly ask for feedback from your team and other stakeholders in your organization. They might not fully understand what your role should be, and they might have little or no prior experience of Scrum to draw on, but they can tell you how they find it working with you, what you're doing that helps them, and what potentially makes their work harder or less effective.
Keep in mind your job isn't to make people happy (although if you're lucky, happiness is a nice side-effect), so try not to view this feedback in simplistic terms, but consider whether it's the sort of thing people would say to a good Scrum Master.