I thought I would provide a trip report on our recent trip, since we've used this forum extensively to help plan our trip. It's very long (continued into the comments section) but we figured it could help with other travelers itinerary.
Berlin: We stayed in Circus Hotel for 4 nights. The location/area of the hotel is great, it's next to a subway/tram allowing for easy access to most sites if you don't want to walk. I would highly recommend the hotel but make sure you ask for the inner courtyard rooms as the ones facing the street are very loud since we had to leave the windows open since there's no AC. We arrived in Berlin (25 minute taxi ride, 25 euro to hotel from airport) on a Saturday around 11am, we ate lunch at the hotel cafe then went on the Insiders tour (overview) which was 4 hours. It was a great tour, we learned a lot in those 4 hours and it flew by. Getting around Berlin is very easy with their trams and subway. Day 2: we visited a bunch of museums we were interested along with the Berlin Wall Memorial, which we definitely recommend. We visited Mauerpark where they seem to have a music festival on Sundays. It gave off a great vibe everyone was enjoying themselves, everyone from young children to older adults. We had dinner in the Prenzlauer Burg area, which was a nice area to explore. Day 3: Reichstag Building (definitely make reservations especially for the guided tours) and explored the Kreuzberg neighborhood. Day 4: Insiders tour to the former Sachsenhausen Concentration Camp. The tour guide did a great job explaining the camp and we both learned a lot about the memorial. The last day we visited the German History Museum which were pleasantly surprised with. We could have spend hours there. The food in Berlin is great, it is definitely an international city with all the different food options. Berlin opened our eyes that Germany is not all the stereotypical German type food/beer gardens, which is more in Bavaria/Munich. If you want more info on were we ate let us know. We thought the feel of East Berlin was like Williamsburg, Brooklyn but even better. We loved just walking and wandering the streets in this area. We both agreed for being a big city it was cheap compared to other cities. Overall we loved Berlin, it's a colorful city with a great vibe to it. At minimum you need 3 full days, I would say 4 to 5 would be better.
Prague: we traveled by train, when you book a train make sure to look at both Czech railways and DBahn. Czech railway ended up being almost half the cost. It was nice train ride and took about 4 1/2 hours. We had 2 full days in Prague so it was a packed 2 full days but very enjoyable. We stayed at Hotel Hastal, which was a great hotel with breakfast, AC, in a great location, and quiet. We went on a tour with Personal Prague Guides with Lenka. This was a great investment because she brought life to all the sites and easily got us around the city. We also learned a lot about the culture and the history of Czech Republic. That night we went on the Urban Adventures beer and tapas tour. We were actually surprised at how good the tour was! Our tour guide Travis has experience with brewing and brought us to a couple small bars with very good local beers, it was definitely one of the highlights of our trip. He also explained to us the history and culture of Czech Republic as well. We personally liked Czech pilsner/pale lager beers more than German beers. Day 2 in Prague we explored what we didn't see with our guide. One great off the beaten path cafe the tour guide recommended to us was Kavarna Novy Svet. It was on a beautiful side street, I think my wife probably took about 100 pictures of the street. We then went to a small microbrewery called Pivovar U Dobrenskych, which was recommended by our beer tour guide. They had very good beer/homemade food. We were also astonished of how cheap it was. For an appetizer, 5 beers, 2 entrees, 1 dessert and tip it was 45 euros!