You should visit the Ateljee bar at Hotel Torni, in addition to drinks they have a great view of Helsinki from the ladies’ room 😉
If the beer tram (Spårakoff) is running you could take a ride, that should be fun.
I always go to the same restaurants:
Bar 9 on Uudenmaankatu (nice relaxed atmosphere and food)
Tori on Fredrikintori, similar to Bar 9, nice and relaxed
Bali Hai on Iso Roobertinkatu, NOT Indonesian despite the name, also quite similar to Bar 9 and Tori, trendy and good, simple food.
Suomenlinna fortress island is nice if the weather is good, it’s also a fun ferry ride from the market square. You should also take a stroll at the market, there’s some handicraft and souvenirs for sale (most of it is pretty bad, but there are some nice stands).
I prefer Café Carusel to Café Ursula, but this is just a personal thing 😉 They’re both along the waterfront close to Kaivopuisto park and it’s a very nice view of the islands outside Helsinki.
Re Finnish design I also like Artek, they sell Alvar Aalto stuff, mostly furniture though. If you’re interested in Finnish stuff you should check out Marimekko and Iittala in addition to Design Forum.
If you’re interested in general shopping, you can’t go wrong with the Stockmann department store and it’s book shop, Akateeminen, they have a pretty good selection of most things.