To get to the nearby German island Sylt (In Danish; Sild), you have to take the ferry. It leaves from Havneby on the southern end of Rømø, and arrives at the town of List on Sylt.
It leaves about every 1½ hour, and the trip takes 30 minutes. It is possible to bring your car, but I think the best way to explore Sylt is by bicycle.
For a return ticket the fare is 35 DKK in low season and 49 DKK in high season. To bring a bike it is 43 DKK and a car is 418 DKK including passengers.
On the ferry you can shop, eat in the restaurant or watch out for seals on the sand banks in the water.
The website is only in Danish and German, but it should be easy to find prices and the timetable, though.
The small Rømø island is located in Wadden sea on the edge of Denmark and very close to Germany. So I suggest if you are visiting North Germany maybe you can catch it.
Rømø with main land is connected by dam which can be easily be crossed by car and there is no special fee to be paid. I think the driving with bicycles are also available through it.
Coming to the island you can see the island and definitely its beaches. And if you get lucky you can also see many sheep there doing funny things.