Oh, just about anything that's 17th-19th century - buildings, artwork, parks, history.
Also interested in theater, music, and dance; again, the more classical, the better.
Jim is not interested in ancient history or modern art... but I am; I can have a day or an afternoon to myself, too.
ETA: Jim wants me to say that he also likes some 20th C art (e.g., Mucha) and is interested in some 20th C events (e.g., we had a fascinating visit to the House of Terror in Budapest).
Stockholm will be fun for you then. Take a long walk in the old city (Gamla stan) and that should keep you occupied! Then there are some great museums in Stockholm: armémuseet (army museum), medelhavsmuseet (Mediterranean museum), Vin och sprithistoriska museet (wine and spirits museum) and the royal textile museum in the cellars of the royal palace are all world class museums IMO. There are dozens of small, specialist museums in and around the old city.
And for some great beer and "classic" Scandinavian pub food, go to Akkurat and order their strömming (herring). I can't help with wine stuff in Stockholm, sorry!
Helsinki, sadly, won't be as interesting (though if you do come here, we should meet up for a glass or three!). The only must see destination I can think of is the Suomenlinna fortress island just outside of Helsinki (ferries go there several times an hour so it's easy to visit). And if you want something natural instead of museums, the Nuuksio national park is within the borders of greater Helsinki and is also easily reachable by public transport and might be of interest if you don't have experience with northern forests and lakes!
If you like architecture, Alvar Aalto designed many buildings here (I'm not too fond of him, however!). The classical music scene here is really good and I'm pretty sure I can find some good concerts or opera or ballet for you once you figure out what days you'll be here.
The Atheneum, National Museum and the Amos Anderson museums are ok, but not on the level of Stockholm's museums.
For food, here's my short list of the best: Chef & Sommelier, Farang, Spis and ASK. Farang does Southeast Asian food, the others are all modern Nordic.