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?
You can return to orbit to start the next mission. On Xbox, press the Select / View button (the small button on the left side of the controller, between the Xbox button and the left stick), then hold Y to return to orbit. (On PS4, I think you press the touchpad to bring up the same menu - not sure what button returns you to orbit, but it should be shown on screen).
Once in orbit, there should be a button labeled View Director (I think, it should be in the lower left corner I think). Once in the Director Earth should have a mark indicating you have an active quest there. Select Earth, then select the next mission.