Kamppi is a the area with two shopping centres.
Kallio is the “trendy” area, and lots of 2nd hand clothes shops - UFF is the main one and there are a few branches across the whole of Helsinki.
in terms of restaurants, they are traditionally expensive but they have gradually added some lower price places like the falafel chain Fafas.
museums are not free, but you can get tickets to visit multiple museums/art galleries so works out cheaper if you get that type of pass.
kaisan kafe is worth a visit - they are famous for their HUGE cinnamon buns at decent prices along with other baked goods.
the market place in the main harbour is worth visiting. I like ordering a bag of fresh peas to shell and eat at the waters edge. close to this is the cathedral with the big steps.
As well as the main harbour with the market place, the one near Eira is also beautiful and there is a public sauna round that area too if you fancy trying that out.
There is so much more to the city but that is off the top of my head!!