We did a cycle tour, that took us to Hitler's bunker and also to a brilliant cafe by the canal in the Tiergarten for lunch, plus all the other sights.
We did dinner at the Reichstag as a special treat on our last night (you need to book in advance) - great to watch the sun go down from the restaurant at the top and the best bit was going up the dome at about 10.30 pm after we had finished dinner - it had shut to general public by then, so was empty, and brilliant view over the city at night.
Day trip out on train to Potsdam - lovely little town and great to walk round the various palaces/grounds and see the bridge of spies. We did the tourist bus but hopped on and off and walked quite a lot too, taking a picnic. The train trip out is good, you go past lots of little 'summer houses' outside Berlin itself.
We were a little disappointed in the East Side gallery.
Stasi museum (not the prison, we didn't have time to get there) was excellent - out in the East side, we spent half a day there, I could have spent longer. Very sinister feeling to the whole area around there, as it used to be Stasi housing, and fascinating exhibits, including how the government used various surveillance methods. Some of the rooms have been kept exactly as they were.
Day trip out to Sachsenhausen concentration camp - again take the train, then walk, takes the whole day and is emotionally exhausting - don't miss the medical bit near the end (even though it is very harrowing). Just the walk out to the camp from the station is very sobering and makes you really think how these awful places were so near to 'normal' villages and towns and yet everyone was ignoring what was going on inside them. This camp was one of the prototypes and continued to be used after WW2 for other prisoners.
Visit a market (we were staying near Winterfeldstrasse in the west side and there is a good market near there on a Saturday and at least one weekday).
There is a lovely street full of Bohemian cafes and shops and lovely houses which is good for a mooch and some people watching - I will try and find a map and work out where it is
Climb up the dome of the cathdral and do the whole circuit of the roof. Also see the crypt where all the emperors and their children are buried.
We wanted to go up the Television tower and ran out of time - it is supposed to be good to have breakfast up there. We also never made it to the Tempelhofer park which is on the old airport site, from where they did the Berlin Airlift.
Visit KaDeWe (Berlin equivalent of Harrods) for the Food Hall - you can try loads of the food and get given free fabric shopping bags to take your purchases away. Good place for lunch or early supper - just go and sit at all the different counters where you can buy food to eat there.
I always try to read novels set in the city that I am visiting whilst I am there: I can recommend:
Alone in Berlin
The Innocent
Book of Clouds
all set at slightly different time periods within the last 70 years or so.