I got Destiny and I did the first mission where the guy wakes up and has to get to his ship. Now I'm stuck in the tower. There are no mission notifications or anything. How can I get the next mission and start it?
Sounds like you are in the Tower or a similar social area that you go to between each mission (if my memory serves me correctly, it has been a while). To get to the next mission you need to get to orbit.
I played on PS4, but according to this screenshot it looks like the xbox one controls are to push and hold "Y" to go to orbit.