1) For a middle school student usually means walking to classes together, giving hugs, dancing to slow songs at dances, possibly going on "group date" on weekends, and communicating regularly.
2) For a high school student usually means doing that above^ and usually more one-on-one dating on a regular basis.
1) "Scott and Beth broke up after four months of going out."
2) "I saw Jane and Mark at the movie theater togther. Are they going out?"