We have done this cruise 3 times and found something new each time. We had exchange students several years ago. There was one from Sweden, one from Denmark, and one from Finland. Our first tours in those countries was a private guided tour by the students. For your first time there our recommendation would be to do your own walking tour of the old town in Stockholm. The ship docks within walking distance from the town. Royal Caribbean provided a shuttle bus but the walk along the waterside was more interesting. We ended up spending the entire day looking in shops and snacking on interesting food.
In Copenhagen the ship docks close to the little mermaid park and from there it is a short walk to the city center and lots of shopping. The parks and the shops / restaurants along the canals are a must see. They are also close to the government center where you can watch the changing of the guards at noon. Everything in Copenhagen is very expensive and with the dollar exchange rate it is more expensive now.
In Helsinki the ship docks fairly far from the city center. There is a shuttle bus for a fee which is a lot less expensive that a cab. There is so much to see and do in the city that you will not be able to see it all in one visit. Get the "what to see" map from the cruise ship and be prepared to spend an entire day looking around.
In St. Petersburg we contracted for a private tour with DenRus. They provided the visas, a tour guide, and a driver. We got 6 others to join us so the two day tour cost less than the half day tour offered by the ship. You have to see the Hermitage, both of the summer palaces, The church of the blood, and the winter palace.
We have posted pictures of two of our cruises to the Baltic at
view link along with other albums. If you have any specific questions just ask.