Hakaniemi or Hakis like we Helsinkians call it. This truly is the old communist area of Helsinki. Communism might be a swearword for some people, but take it here as a good thing. It's basically because Kallio used to be workers part of the city, so Hakaniemi was kind of it's centre. From Hakaniemi you'll find the best marketplace for vegetables during summers as well as market hall which runs a year around and where you'll get the freshest fish and vegetables grown biodynamically. Then there's also Comedy theatre, a lot's of Asian supermarkets, access to Merihaka. Also Tokoinranta is a good place to spend time with locals.
Also restaurantwise Hakaniemi is a paradise. You'll nice fine dining style restaurant Kolm3nen, all vegetarian restaurant Silvoplee and best bouillabaisse in Helsinki. And how about trying original Karelian pie - Hakaniemi market hall is the place for them. Just to mention a few. Hakaniemi is a compact area where you'll find everything and actually if you don't want, you can just stay in that area for a week without going to anywhere else. Also there are many places for children to play.
And psst. you'll find also one of the best bars in Helsinki from Hakaniemi too.