When you are on Departure (or Approach) ATC, and you are planing to leave the airspace you can request flight following to your destination airport.
Some time ago Mark made a good tutorial on this, you may wanna review this.
Apart from the tips in the tutorial, there’s one other thing I would like to ask pilots, on behalf of all Approach controllers: You don’t have to do a ‘check-in’, prior to requesting flight following. So please don’t. Just request flight following to your destination airport and fly your flight plan. That’s it. No check-in needed first.