You could cross with a visa and travel permit. Most non-German auto traffic traveled along the Hanover–Berlin autobahn (Bundesautobahn 2). Crossing from West Germany into East Germany at Checkpoint Alpha, where papers checked and cars searched by border guards from both sides. US citizens, at least, were briefed by US military personnel on how to act and what to do when crossing, and the British and French also had people there, as well as the Soviets. At the entrance into West Berlin, Checkpoint Bravo, cars were again searched. Cars were logged in and out, so if you deviated from the expected time, you would be questioned. The return trip was the same. Both Alpha and Brave were fairly extensive facilities.
West Germans had a slightly easier time, but still had to obtain travel permits. East Germans were much more restricted. There was some freight traffic between the countries as well. For this inner-German traffic, there were more border crossings available.