Visit Uspenski Cathedral near South Harbor. The largest Orthodox church in Western Europe and it is lavish on the inside. It dates back to 1868 and is free to visit.
Consider taking a free walking tour to visit Senate Square, the Helsinki Cathedral, and the South Harbor/Market Square. It is a great way to take in the sights.
Take a ferry to Suomenlinna, a UNESCO World Heritage Site, and a popular half-day trip from Helsinki. Suomenlinna is made up of small island fortresses with cannons.
Embrace Finnish sauna culture. A couple of popular public saunas in Helsinki are the Löyly and the Allas Sea Pool. Enjoy outdoor swimming pools including one that is fed by the Baltic Sea.
Take lunch at Vanha Kauppahalli, a traditional Finnish market from the 1800s. Try delicious Finnish food including fresh salmon soup and tippaleipä which is like a funnel cake.
Museums
The National Museum of Finland covers Finnish history from prehistoric times through the 19th century. The Design Museum is a museum in honor of Finnish design.
Go to Esplanadi Park for live music or consider a stroll through Sibelius Park, a forested park which is home to one of the most charming cafes in Helsinki, Cafe Regatta.
I recommend visiting more of the islands in the Helsinki archipelago by booking an afternoon sightseeing cruise with commentary or an evening sunset cruise.