Central Helsinki
I'm a bookworm, so I start with libraries. Oodi attracts different kinds of people as its opening hours are very convenient. If you would like to go to library with a bit of history, I'd recommend the Rikhardinkatu library. Helsinki University library Kaisa is also worth a visit. Rosebud bookstore is also in the vicinity.
Kauppatori (Market Square) and Kauppahalli (Market Hall) are worth checking out if you want to experince torikahvit (buy a cup of coffee and just sit down and watch what happens around you).
From Market Square you can take the ferry to Suomenlinna (there's also a small library!) and have a stroll around the sea fortress. Korkeasaari Island is actually a zoo, accessible via ferry during the summer months, from Railway square you can also take a bus 16 all year round.
The Ateneum Art Museum is so near the Helsinki Railway Station, you can't miss it!
As for restaurants I'd recommend
Finnjävel
Magu (vegan restaurant)
Ravintola Lasipalatsi
Aleksanderplats
Further from the city center
Even though the Helsinki City tourist maps usually don't cover Eastern Helsinki, here's a few tips:
Itäkeskus for shopping
Uutela for nature (take the metro to Vuosaari and along Aurinkoranta)
Alppiruusupuisto (take a train to Huopalahti) & Kirsikkapuisto (walk from Siilitie metro station): these are of course open all year round but rhododendron and cherry trees blossom in June and May.
