When I visited Berlin's Museum Island ten years ago, I had no idea it was a World Heritage Site. Consequently, I only visited the most celebrated museum on the island -- the Pergamon Museum. The reconstructed Ishtar Gate was as spectacular as I'd expected, with its brilliant blue brickwork bedecked with aurochs and dragons. I also appreciated the Pergamon Altar, with its exquisite friezes of Greek gods. I was especially impressed with how large both the Ishtar Gate and the Pergamon Altar were in person. Germany acquired these works of art through archaeological expeditions in the late nineteenth and early twentieth centuries. I understand in today's world that it may be better to keep such cultural treasures in their home countries where they can be seen in context, but these ancient works of art and architecture that made their way to Europe in an earlier era are well-preserved, and have at least inspired me (and hopefully many others) to travel more and learn more about ancient cultures.
Logistics: Museum Island can be reached by bus, taxi, or a walk from a U-bahn or S-bahn station. While on the island, the view of the city and museums from the top of Berlin Cathedral (Berliner Dom) is amazing.