Obviously I come here with my own car. But if you are a tourist and need to use public transport it is not that easy but still it is possible. From Helsinki you take the commuter train to Espoo center and from there bus no. 85 and ask the bus driver to tell you when to get off for Nuuksio park. Its a short ride, maybe about 15 minutes.
It pretty easy to reach Nuuksio from hte greater area of Helsinki.
If you come by car you will most probably want to drive on interstate #1 and leave at Nupuri. Here you follow the signs to Haukkalampi and Nuuksio... until you reach Haukkalampi parking area.
Your options to arrive by bus are quite good. You can take either 28T or 28A or 85 or 85A leaving from Espoon keskus.