Although some county officials ask for the address of your destination(s), you are not obligated by Nebraska law to share that information with your resident county.
If you travel within Nebraska, you must notify your local county of your “temporary domicile.” In Nebraska, a temporary domicile means any place at which the person stays for a period of at least three consecutive working days.
So, if you’re traveling from Omaha (Douglas County) to grandma’s house in Grand Island (Hall County) for a week, you must notify the Sheriff's Office in both counties.
As a Nebraska registrant, if you travel to another state, it is your responsibility to know what that state’s legal requirements are; otherwise you could be facing a felony. Some states’ requirements are statewide, and others are by county and city. In some states you could have 3-10 days before you must register the location at which you are staying, and in some areas it could be within the first 24 hours.
Each state is different, and fully-researching the laws of your destination is imperative. Not only may you be required to follow their local registrant laws, you could be required to register. For example, if you visit the same Florida location for five or more consecutive days, or any place within the state for 30 days aggregate during any calendar year, you must register. Then, when you return home, you remain on the Florida registry. For life (regardless of the duration of your Nebraska registry).
Traveling internationally is a bit stickier, but information about your destination country and its requirements may be found here: http://registranttag.org/resources/travel-matrix/
No matter your destination, if your trip required you to report it, you must contact your county sheriff when you return.
Contacting both your local registration office and the registration office in the state/county you are visiting, even when traveling intrastate, may save you some trouble. A phone call to your destination's local authorities is recommended, ensuring you know how they interpret/enforce the local laws. Often this will be a fairly informal process, but the burden falls on you to understand and follow the registration requirements.
This information is not legal advice, and should not be considered as such. Please check the laws of your originating and destination cities every time you travel, as laws may change.