1
1. Old Town.2. The easiest day trip is Helsinki. The trip through the frozen sea is worthwhile in itself.
3. Book your restaurant ahead for New Year's Eve.
2
As @#1 points out, the boat trip to Helsinki is really nice. The sea won't be frozen, don't be alarmed.5
Just outside the old town is the seaside town of Pirita. I know it is hardly the weather for the beach but the road between the two has the excellent Kadrioru Park with its palace.And the truly awesome? weird? soviet war memorial.
Also go into the soviet style building of the local library, just beyond the city walls
