We flew into Copenhagen the day before the cruise, and easily collected our bags and navigated to the metro station. The airport is HUGE, but really well laid out and nearly all signs are in English. The Danish are also a really nice group of people and we found them more than willing to help us during our stay.
We stayed at the "Hotel Bethel" in Nyhavn (the part of town with the cute, brightly colored row houses). The rooms were tiny but sufficient, and the breakfast was great. The staff was fantastic and super helpful. I found it to be an excellent location, though honestly I don't think there's a bad location in the whole city. The city (at least, all the parts you'd want to visit) is very compact and completely walkable. We walked back and forth across all the touristy sections and then some at least 7-8 times during our time there. We also spent a day there post cruise. Below are the things we did in town.
*Boat tour - This was entirely skippable, though it was a nice, relaxed way to get a feel for the city the first night when we were so jet lagged and tired. There are 2 booths selling tickets, one for 80 DKK and one for 40 DKK. I don't know that there's any difference (our hotel concierge said there wasn't, other than price), so we took the 40 DKK one and it was fine.
*Ice Bar - Don't buy tickets in advance (though a reservation is a good idea). At the end of Nyhavn, where the canal stops, there was a guy with a table set up selling discount tickets to everything. For the same price as you could pay at the door, his tickets included 2 drinks instead of one. At $30pp, it's a pricey little tourist trap, but we had fun. The drinks were STRONG. We wandered over there around 6:30 and there was only one other couple for 45 minutes until a group came in.
*Tivoli - We opted not to pay and go in. It was around $17pp just to walk through the gates, and then more to ride anything or eat there. We walked the fence line and peered in and got a feel for it, but opted to save our money.
*Little Mermaid statue - Everyone said this was "so far away", but it really wasn't. It was a very pleasant 20-30 minute walk from Nyhavn through an old fortress and a park along the waterfront.
*Sandeman's Free Walking Tour - EXCELLENT. Must must must do if you are up for walking. It's about 2.5 hours long, but there's a "coffee break" in the middle of it, and the guide did a fabulous job of finding places for us to sit whenever he was going to talk for a while. He was really knowledgeable and funny and earned ever penny we gave him in tips.
We didn't have an interest in going to Berlin, so we hopped on a train and headed up to Schwerin for the day to see the castle. It was a good 90 minutes of travel for a town that really wasn't all that much. The castle was pretty, but it was hard to get much out of it without speaking/reading German. There was some signage in English, but it would have been really helpful to have had an English tour of it! Others on our ship had good things to say about the "Friends of Dave" tour--next time I think I'd do that instead.
What a pleasant surprise! We followed the walking tour that is so widely recommended on the boards here. There was construction that interrupted a stop or two, but in general it was pretty good. We found a spot in the wall near where they sell woolens on the perimeter where you could go up and walk along the top of the wall which was pretty cool. Again, we probably would have gotten a lot more out of a tour, but hindsight's 20/20
We did the 2 day "As much as physically possible" tour from TJ. Our guide was Maria, who knew everything about everything! The favorite spot was definitely Peterhoff and it's gorgeous fountains. What a neat place! I picked TJ because I wanted authentic Russian lunches instead of sandwiches like the other popular company on our roll call offered, and we weren't disappointed. We had 3 course lunches (soup, entree, dessert with tea/coffee) with Chicken Kiev one day and Beef Stroganoff the next. I can't say I was a fan of all the food, but I'm glad we got the chance to try it. My only complaints about the trip is that it was very fact-based, rather than the guide weaving in stories from her daily life, which is often my favorite part. The other thing is that the store we went to in order to pay for the tour and buy souveniers was EXPENSIVE. Maria told us it had good prices, so maybe everything in Russia is really expensive, I don't know. We bought an egg Christmas ornament that was around $30.
This was a throw-away port. We slept in and got off the ship late (around 9) and walked all the way into town, which was about an hour. Apparently there was a bus to take for 2.7 Euros, but I didn't pay enough attention. It was a really pretty walk along the water though, around 3 miles. We took the tourist tram line around the city (bought a ticket from the driver) and got off at the rock church. Got back on and tried to find the Olympic stadium, but found an international youth soccer tournament instead where we found a team from our home town! Took the tram back to Market Square and ate the most amazing apricots and strawberries for lunch, before walking back to the ship. My legs were exhausted at the end of that day!
Disappointed that we had such a short time here, but I knew that going in and honestly I was so tired of walking at that point it wasn't that big a deal. We were supposed to be in port from 7-2:30, but when the daily paper came out it said 8-2:30. We raced off the ship at 8:00 and ran to booth right off the gangplank to buy HoHo tickets. It got us to the Vasa Museum at 8:30, right when it opened. We had downloaded the ipod tour and wandered around with that for 30 minutes until the 9:00 English showing of the movie. Got out of the movie to find the cruise tours had arrived and the museum was pretty packed. I was about ready to leave anyway, so we finished up the ipod tour and got on the HoHo boat again at 9:45 or so to go to Gamla Stan. They should rename it "overpriced souvenier stan". There wasn't really anything there other than tourist shops. We found a 7-11 where we bought sandwiches and just wandered around until noon for the changing of the guards ceremony at the palace. It was quite the event! The Naval Cadet Band played all sorts of music, the guards did a bit of running around, and hundreds of tourists stood in the pouring rain watching it all! We then took the HoHo boat back to the ship and got back on around 1:30. If I could do it again, I think I'd go to the Nobel museum (in Gamla Stan) while waiting for the guards.