On a flight with short stops, do you terminate flight following at each stop?
As @RonBeyer notes in the comments Flight Following will drop you when you land. Generally (also as Ron notes) they will ask you to call the "field in sight". It really depends on where you are and what kind of air space you are in. Where I fly in the Northeast they typically drop you when you have the field in sight but its hard to stray to far from a heavily controlled area if you are flying between DC and Boston but it may be different elsewhere.

Keep in mind that you will need to get off the frequency so you can make your CTAF announcements anyway. It may be a good idea to keep the controller as a listen only channel on COM2 if you are equipped for such in your aircraft. 

Assigning you a code is not really a hassle for them so you don't need to worry. In a busy airspace ATC may want you out so they can handle IFR traffic. Remember flight following is "load dependent" so you might not always get it and you can be dropped out at any time. 

When on your student XC's its always a good idea to preface one of your initial calls with "N123AB Student Pilot requesting flight following". ATC is there to help you and noting you are a student pilot may help you so they don't expect professional pilot level flying and radio chatter from you.