For reasonably priced but good restaurants in Helsinki I would the recommend following.
Kuurna http://www.kuurna.fi/ (a personal favorite)
Olo http://olo-ravintola.fi/ (probably the most expensive of the bunch but still not on the 2 michelin star Chez Dominique level)
Atelje Finne http://ateljefinne.fi/
Chef et sommelier http://chefetsommelier.fi/ (check http://www.theatlantic.com/international/archive/2012/05/can-one-chef-take-finnish-food-from-worst-in-the-world-to-gourmet/257209/#.T7XfqpW5gbV.twitter)
These are mostly scandinavian cuisine with varying amounts of french or italian. There are plenty more, but this a good start that should help you to keep out of those horrible chain restaurants dominatin the city.
For Stockholm I don't wan't to recommend any because I have been only to few and I know there lots of swedes on the forum too.
Hope these help and you haven't already done yor trip.