If yours is a museum-heavy itinerary (likely, given Finland's winter weather), it's possible the Finnish museum card would save you money. It's valid for a year, and I found it covered a very large percentage of the museums I wanted to see. You'd need to check entry prices to see whether it would be financially advantageous; it's an incredible bargain for museum lovers spending a good bit of time in the country. https://museot.fi/museumcard
My pre-trip notes for Helsinki follow. They focus on art and Art Nouveau architecture (of which Helsinki has an impressive amount). They're organized more by geography than by priority. I didn't manage to see all these places.
- Didrichsen Art Museum: Finnish and international 20th-century art. Also pre-Columbian and Oriental art.
- Villa Gyllenberg Art & Home Museum, Kuusisaarenpolku 11: Collection includes contemporary art.
- Kapyla: Laidback Bohemian neighborhood with colorful, old wooden houses in area W of Kullervonkatu and north from Koskelantie to Pohjolankatu
- Iittala & Arabia Design Centre, Hameentie 135: Lab & Design Museum Arabia (8th fl.), shop (2nd fl.) and cafe.
- Toukola: Idyllic neighborhood with wooden houses.
- Vallila: area with wooden Art Nouveau buildings. Easy to reach via Tram 1 or 7, stop Vallilan Kirjasto.
- Talvipuutarha (Wintergarden), Hammarskjoldintie 1: Large greenhouse with tables where one can picnic.
- Sibelius Monument
- Toolo: Known for beautiful 1920s architecture
- Reitz Foundation Museum, Appolonkatu 23: Furnished period home with art, silver, porcelain.
- Kirpila Art Collection, Pohjoinen Hesperiankatu 7: Large collection of Finnish art 1850s – 1970s.
- Kansallismuseo (National Museum of Finland)
- Villa Hakasalmi: Check on what sort of special exhibition is running.
- Kiasma Museum of Contemporary Art: International art 1960 to present.
- Central Railway Station: Architcturally important (Saarinen).
- Amos Rex: International contemporary art. Temporary exhibitions only, so check website to be sure what's on interests you. Museum may be closed for quite a while as it sets up a new exhibition.
- HAM (Helsinki Art Museum): Contemporary art.
- Kunsthalle Helsinki: Art
- Temppeliaukio Church (Church in the Rock): This is a quick visit.
- Ateneum: Largest collection of art in Finland, including local favorites plus works by Cézanne, Gauguin, Van Gogh.
- Suomen valokuvataiteen museo K1: New branch of Finnish Museum of Photography.
- Senaatintori (Senate Square): One source called it "Europe’s finest Neoclassical square".
- Lutheran Cathedral: 19th-century neoclassical masterpiece.
- Kruunuhaka: Residential district with tiny streets; architecturally interesting.
- Uspenski Orthodox Cathedral: Byzantine-Slavonic style.
- Katajanokka: Area with elegant Art Nouveau buildings, some of stone.
- Arkkitehtuurismuseo (Museum of Finnish Architecture)
- Design Museum, Korkeavuorenkatu 23: Chronological exhibition plus cutting-edge temporary exhibits.
- Sinebrychoff Art Museum: Art from many private collections.
- Tehtaankatu and perpendicular streets to right: Art Nouveau District.
- Huvilakatu: Art Nouveau buildings, plus some on surrounding streets.
- Eira: architecturally interesting neighborhood, including Art Nouveau villas.
- Suomen Valokuvataiteen Museo, (Finnish Museum of Photography), Kaapelitehdas (Cable Factory), Kaapeliaukio 3
I didn't have time for Suomenlinna, the sea-fortress island, and I have no idea whether things out there are open in the winter.
I found the Helsinki City Museum not particularly interesting: It's popular with local families.
Running out of room so I'll do a separate post for Turku.