Copenhagen + Hamburg - seeking recommendations

I can't speak to Hamburg, having never been there, but we were in Copenhagen in late September/early October last year. I think it is a wonderful time to visit, and temperatures should still be relatively comfortable (we had low 60s just about every day).

If at all possible, try to fly open jaw. Perhaps your husband could ask his employer? It might even be cheaper for the employer, Copenhagen has a lot of nonstop flights to the US.

I think four nights is a good amount of time in Copenhagen. We spent seven there, but did a lot of day trips. We didn't go to Malmo either, in part because of also reading that there didn't seem to be anything special to do/see, but also because there were so many places to go and things we wanted to do in Denmark.

I would prioritize seeing the Rosenborg Castle and Gardens, Christiania, doing a canal tour, and the National Museum (excellent). There is a ton to do in Copenhagen, so I'd suggest picking up Rick's book and diving in deeper. What interests you? Art? History? Shopping? With four full days there, you also have time to do some day trips. Popular ones include Roskilde (Cathedral where all Danish royalty buried and a UNESCO World Heritage site, also an excellent viking ship museum); Helsingor (Kronborg Castle (Hamlet), a nice coastal town, and a great maritime museum); and Hillerod (Frederiksborg Castle). Also north of Copenhagen (and on the same train line as Helsingor) is the stunning Louisiana Museum for Modern Art in Humlebaek.

The best restaurant we ate at was Relae, which was recently ranked as the 40th best restaurant in the world. They have a four course prix fixe for about $70 USD per person (not including alcohol) and it was spectacular. There are other high end, world-renown restaurants in Copenhagen (Noma, Geranium, etc.) that require reservations way in advance and are very expensive, too rich for our blood, but would be exceptional experiences. The Cofoco restaurants are a great place to try out Danish cuisine at (by Danish standards) a reasonable pricetag. We ate at Cofoco and Spuntino (they have a few other restaurants) and 4 or 5 course meals were excellent and about $45-50 per person. For a fun afternoon, go to Copenhagen Street Food on Papiroen (an island near Christiania). It is a big food hall with practically every type of ethnic cuisine you can imagine, at very reasonable prices ($10-15 USD for a very filling meal). They also have outdoor seating right on a canal, and it's a wonderful day to pass some time and people watch. Definitely get to some bakeries for breakfast. We tried to go to a different one every day, and they are everywhere.

What else is there? Do you have specific questions? If you plan to go outside of Copenhagen (such as for some of the day trips I suggested), consider purchasing a Copenhagen Card. It includes free admission to many places of interest as well as free public transportation pretty much everywhere on the island of Zealand. You'd need to figure out whether it is worth it for you to buy it.

We were in Copenhagen last August. Great city and easily navigable. We opted to get the Copenhagen card for unlimited bus, train and discounts or free entrance to attractions. We stayed at the Hotel Alexandra http://hotelalexandra.dk/ . Super location near train station and is decorated in Danish modern (50-70's) furniture. Each room is furnished in a year. (our's even had a fashion magazine from the time as reading material) Plus they are very "green" and will give you a voucher for food or drink if you elect not to have your room cleaned and reuse your towels.
We hit a lot in the short time we were there. Canal cruise, Christiania, Carlsburg brewery tour. We did enjoy a morning to Roskilde cathedral and the Viking Ship Museum. (great food at the Viking experience, too) On the same day we headed up to the Louisiana Museum of Modern Art. It is a beautiful modern art museum with great views and awesome sculptures!

If you stay at the hotel Alexandra there is an Italian restaurant located on the same block behind the hotel that was absolutely brilliant.

This hotel is just a block from Tivoli gardens, I am not sure if it was free with the card or heavily discounted. We went one evening walked around a bit, but it was crowded. The entertainment was interesting and I understand the restaurants are good, but didn't try any.
Just wandering around and going in any church is an experience. Some very wonderful picture moments all around. Be aware of the bicycles and don't cross the street until the light says walk.

Sarah:
A few thoughts on Hamburg. My wife and I visited there twice, the last time in 2014. The canal boat trip is a great way to get a sense of the historic (and current) linkage of the city with maritime industries. We took a late afternoon “cruise” and with a couple glasses of wine (purchased onboard); it was most enjoyable as the sun was setting and lights were coming on in the city.

The U-Bahn is an easy and inexpensive way to see various districts within the city. We enjoyed visiting St. Michaelis and taking the lift to the tower observatory level. We were fortunate to be there when a thunderstorm approached the city; we remained outside when others headed for the lift and street level—it was great to have the observation level to ourselves. While there along with walking through the area we visited Peterstrasse (not crowded with tourists), and stopped for coffee with locals . We enjoyed visiting The Old Elbe Tunnel (as pedestrians), the elevators are really something. St. Nicholas’ Church is another interesting site, it remains as a ruin and memorial of the bombing of Hamburg in WWII, and you can take a lift up in the tower. A tour of the interior of the Rathaus is very interesting (check schedule for tours in English), and while there have lunch in the restaurant on the lower level (good food and reasonably priced) You might also consider a day trip to Lübeck.

While in Hamburg we stayed at the NH Hamburg Mitte, Schäferkampsallee, a short walk from the Christuskirche U-Bahn station. It was comfortable and offers a wonderful breakfast buffet.

Thanks so much for the comments and suggestions, everyone! Plans have changed slightly due to schedules and airfare, and with my partner's employer agreeing to open-jaw, I am now flying in to Copenhagen and out of Stockholm! He will have a day (and maybe some evenings after work) on his own in Hamburg to explore, and will use the suggestions you have made to my two posts. He'll take the train up to Copenhagen to meet me and is looking forward to it!