memorial/museum/historical site at Bernauer Straße, where you can see a section of the wall as it would have looked, watch (free!) documentaries about the wall, and gain a sense of what the entirety of the wall was like.
What I didn't understand for a long time, was that the Berlin Wall was actually around WEST Berlin, not East. It isolated West Berlin as a capitalist island in the middle of East Germany. The wall that faced "inward," to West Berlin, was eventually covered with graffiti. West Berliners could get right up to the wall and touch it. On the other side, the side facing East Germany, there was a second wall, guard towers, an electrified fence, and guards...no getting close to the wall. So that side was pure white. On the Bernauer Straße site, you can see this system, and they have a (also free!) tower to climb up and look over/into the "death strip" between the two walls.