We recently tried the pedal cars in Helsinki Traffic Park (Lasten Liikennekaupunki), and I thought the place was brilliant. Families can for free go there on Fridays and Sundays 10am-4pm and learn the traffic rules when pedalling a car. Mon-Thurs there are lessons for diverse groups but after hours and on Saturdays one can also go there with own vehicles and helmets. My kids even loved running around as pedestrians. There is also a fenced off little playground in the middle of the traffic park, so smaller siblings can be close by and watched while training with bigger kids on the streets.
During open hours they also have a tiny café there where everything cost 0,50-3e. It only sells coffee, tea, juice and some snacks. Cash only.
The traffic city is next to Aurora Hospital in Auroranportti 2.
Edit July 16th 2016: Now they also have pedal cars that are easy for parents to push, so that younger kids or kids with poorer coorniation skills can have a facilitated go too.