I would recommend taking the ferry over to Suomenlinna and exploring there to be honest - takes about 15 minutes or so and is part of the HSL system.
Otherwise a trip out to Mellunmaki on the metro to visit the northernmost metro station in the world - not a lot at Mellunmaki; not the best neighbourhood by Finnish standards and not a lot there other than housing. For a quick ride on a FLIRT, maybe a train about to Leppävaara or Espoon Keskus just for the ride - both have large shopping centres (Leppävaara especially) attached; Leppävaara also has the light-rail tram 15 - you could take that a few stops.
If you don't have much time, I'd recommend a walk down to the Cathedrals, over to the harbour, "Vanha Kauppahalli" for some touristy stuff and then the Suomenlinna ferry. Take the tram back to the station on the way back. Try to catch one of the older trams.