Too much to see in 1 day. Lucky for us, we were coming back for a week after the cruise. Must see are the Vasa Museum (the ship), City Hall and walking inside the Gamla Stan. Ice cream shops where they make they're own flavored cone. Also nice to get out of the old town to see the "real Stockholm". The Royal Palace is nice but not a must. Shuttle into town if you want. We walked (20 min).
Took the cruise shuttle into town. Suomenlinna Sea Fortress is a must. It is a beautiful place to walk around and spend some time near the sea. Walked around Helsinki (Helsinki Cathedrale, Esplanade Park, Senate Square, Uspenski Orthodox Cathedral, Old Church Park,...)
You have to take a 2-day tour! Took the cruise line excursion and shouldn't have. Very nice port. Customs: it is exactly like you imagine it. Got to see a lot of places. If you want to see a maximum of things, don't book with the cruise line. Friends of us didn't and got to do a canal cruise, take the metro, see a market. Instead, we had 2 very long lunches with sparkling wine, vodka and typical meal. Was nice but personally, we would have preferred to skip the big meal and see more of St-Petersburg.
Nice to walk around in the old town. Not much to see so it was a relaxing day. A nice way to finish the day: port had an outside bar with music and free wifi. People are not the most friendly, but it didn't bother us.
Loved it! Walked around in the morning (little mermaid, Amalienborg, etc). Went to Rosenborg Castle. Very different from the usual European palaces. Feels like people left a couple of minutes ago and might come back any minute. Loved it! Met Danish friends for lunch. We had a danish hot dog and went in the Nyhavn for some smorrebrod with schnapps. Luxury goods (like restaurants) are really expansive in Denmark, but it was a wonderful experience. Took the canal cruises and walked some more in the afternoon.