First off, it needs to be made clear that there are no "duties" that relate to reputation and privileges. There are a considerable number of high rep users (those over 20K) who do next to nothing in terms of user moderation and that's absolutely fine.
As to how to use your new-found closing powers, there are a few posts that are essential reading (see below), but beyond that it's just a matter of involving yourself in the site.
One of the best ways to learn is to follow the review queuereview queue. This will allow you to see what questions and answers are being flagged by your fellow users. If you're not sure how to deal with a problem that arises, the easiest thing to do is 'skip it' and come back later to see how others dealt with it. You can also look at the history of a contentious question/answer using the timeline tool.
What questions are on-topic, and what questions are off-topic?
What is a "closed" or "duplicate" question, and how does question closing work?
Closing Story-Ident questions as duplicates (where there's no acceptance)
Why is the close-as-duplicate policy so aggressive, and should we change it?
Should we burninate General Reference?
How should we handle questions that are about non-SF/F elements in a SF/F work?