On a cross-country trip, is it normal to terminate FF at each stop and reacquire it (potentially just a few minutes later) for the next leg, or would ATC prefer that you advise them it'll be a short stop and maybe let you keep the code to reduce their hassle later?
Flight Following is intended to track your flight from time of activation to reaching your destination. It provides for ATC services which include traffic advisories, and emergency services. It is expected to terminate at point of landing.
That said, depending upon workload, some TRACONs will provide flight following which continues through approaches or even touch and goes at an intermediate airport. However this is a workload and controller preference. And it is not intended to support full stop landings, or passenger changes. If you routinely make stops, you might let the controller know you will be back in 3 to 5 minutes, and they may keep your ticket alive, but that is only likely if they are not very busy at that time.
Flight following in Canada is called an en route radar service, which perhaps better describes flight following. In the US, years ago, a request was customarily made for "radar services" which were essentially that of flight following.
In summary, flight following is not intended to carry over "flight to flight" where flights are normally terminated by landings. Think of it as radar services, not necessarily a flight plan with SAR backup.
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.