There are two fantastic starting points for Avengers in modern comics: Bendis' New Avengers, and Hickman's Avengers / New Avengers. The first, beginning in 2005, leads in-to and after the events of Civil War. This is a fantastic series for new readers who want to focus on an individual series, but are still really interested in big events:
The second, Hickman's Avengers / New Avengers, is a longer, heavier, darker and more complex saga. This series may be a little much for new readers, just in scale, but if you're up for it, the story Hickman tells through Avengers, New Avengers, Infinity, and Secret Wars that is by far best I've ever read. I've laid out the full reading order on my Secret Wars article.