Some notes from when we visited.
Copenhagen: Tivoli is a delight in the evening, make sure to stop at the Vaffelbagereit, the ice cream is fabulous. Strøget has become very commercial with big international brand names, the side streets have more interesting shops. Just two streets away from one end of Strøget is Rundetårn (Round Tower) which (25DKK / around $3.50 entrance) was an observatory which now offers fabulous views of the city and you can also have a look at the bells of the church it is attached to. If you have an interest in history the palaces (Rosenberg, Christianborg and Amalienborg) are very interesting with the latter two still in use. Rosenberg is the oldest and arguably the most interesting and also houses the crown jewels. Christianborg is the home of the Danish Parliament but the state rooms are open, you can also see the ruins of the old castle underneath it, the Royal Chapel and the interesting old Royal Theatre. Amalienborg is where the current royal family live and is probably the least interesting, but the nearby Marble Church is quite something. When walking from the port you will pass the Little Mermaid statue and Kastellet, a bastion which is worth the slight detour to walk the ramparts especially as it is on the walk into town (about 15 minute walk from ship to town). Separately you will find branches of Paradis Is around the place, they are a homemade ice cream business with very nice ice creams in a range of flavours. For lunch Ida Davidson offers around 180 kinds of Danish open sandwiches which very nice.
Stockholm: I understand our cruise is not docking in the normal area for cruise ships, instead is docking at Stadsgården which is a short 20 minute walk from Gamla Stan. I wouldn't then get the bus to Vasa, but catch the ferry which goes from close to the Royal Palace and docks at Djurgården a short walk from Vasa. The ferry is part of the public transport system so it pretty cheap (single ticket costs 40SEK (~$4.50), a day ticket costs 100SEK (~$11). Djurgården is also home to Nordic Museum, the Abba Museum, Skansen and Gröna Lund (an amusement park) amongst other things. Incidentally the cafe at Vasa is a little pricey but the food is quite nice.
Gamla Stan is interesting and quirky, off to one side across the bridge on Riddarholmen (very very short walk) is Riddarholm Church which contains the crypts of the majority of the monarchs up to 1950. The shops though are tourist specials in the main with prices to match.
Helsinki: The Rock Church is completely amazing, but it is a fair trip to go to (figure 20-30minutes each way on the tram). An interesting alternative (though not as spectacular) is the Kampii Chapel of Silence in the main shopping area, a delightful oasis of tranquility. We loved Soumenlinna, it was one of our highlights of Helsinki and takes longer than you imagine to see, worth visiting the submarine! The ferry goes from Market Square about every 40 minutes and is part of the city transport system and covered by regular tickets. Market Square is much less interesting than you would imagine and the Cathedral is very austere making it somewhat of a disappointment after the climb. Just to one side of Market Square is the Old Market Hall which has a great selection of places to get lunch and snacks.
Check carefully the value of the Stockholm and Copenhagen cards, we travelled hard and couldn't get them to be reasonable value for money. We found getting a day transport ticket and paying admission made more sense.