Good question. “Get to do something” usually means “have the chance or opportunity to do something.” Examples: I got to meet the president. - I had the opportunity. We got to see the concert for free. - We had the chance. It does not usually mean “start to do something.” For that, we say “begin to do” or “start doing.” So the main meaning is: to have the opportunity or privilege to do something.

Not "start to do something", but yes definitely "have the opportunity to do somthing". It also has the meaning of doing something after a certain period/length of time or after an interruption or wait. Eg. After all these years I finally got to visit Rome. After tidying up, washing the dishes, then putting the kids to bed, I got to sit down and watch TV.

Here's a few examples: You keep talking about how cool it is for someone meet their favourite celebrity, but did you actually get to do it yourself yet? Did you get (a chance) to play on your new computer yet? I didn't get (round) to doing the housework this morning, but hopefully I'll find time for it tomorrow.