You can take bus from Helsinki central bus station in Kamppi (there are many buses per hour, you can also catch in various points in Helsinki but the bus station was more convenient because we wanted to check the alternative routes, pic2 shows the timetable panel in the station).
You can use a single regional ticket (4,50e) or better a regional day ticket (1day 12e, 2d 18e, 3d 24e etc), regional tickets are valid in Helsinki, Espoo, Kauniainen, Vantaa (airport)
As there’s not just one center in Espoo you have to take a different bus, for Tapiola (where museum of modern art is) we took bus #105 but you can also take buses
#106, #109, #110 but it seems they terminate on different parts of Espoo. The ride took about 20’.
We asked the driver to stop at TAPIOLAN KESKUS, then we walked 100m up where there was a mall where we had a coffee break and then 300m up again we visited EMMA(Espoo Museum of Modern Art)
From Helsinki center you have to drive 10km to the west to reach Tapiola in Espoo or some km more of other districts in Espoo
It is much more difficult to move around in Espoo than in Helsinki without a car. There is a good bus network connecting different districts but it gets confusing with so many different residential areas in Espoo. We just took the bus from Helsinki because we wanted to see museum of modern art in Tapiola district, there was a map (pic 2) of the area in the mall but you can easily walk to the museum, there are signs through out the main street (pic 3).
Next day we used a car to reach some parks and lakes outside Espoo.
You can take one of the walking/hiking trails in Nuuksio National park (bus 28 from Leppävaara for southern part or bus 85A from Espoo Centre to Kattila for the eastern or northern parts)
See timetable at http://www.hsl.fi/EN/Pages/default.aspx