If it was summer I could recommed the historical sea fortress '
Suomenlinna' (on an island right in front of Helsinki), might be a bit cold and grim this time of the year though
. Other tips:
Helsinki at ease: Favourite places - Helsinki.fi city portal.
edit: For getting around I'd recommend public transport, a three day ticket is 14€ (see:
http://www.hsl.fi/EN/Pages/default.aspx, the route planner is handy, give it from and to street addresses and it figures out a way to get there with the next available public transport, the 'metro' and the trams are the most convenient with their frequent schedules
).