Just about 2 hours away from Oslo, and the train passes through some amazing scenery. Unfortunately during part of the trip there was a literal whiteout due to snow. But what I saw was really nice ... white winter landscape ... gorgeous :-)
Trains run from Oslo to Lillehammer frequently and the trip takes 1-1,5 hour, ideal for a day trip.
I was travelling with the Eurail Pass, so I don't know exactly how much does the ticket cost, but I presume it is not expensive.
An old paddle steamer plies Lake Mjøsa during the summer. In fact, "Skibladner" the oldest still-running paddle steamer in the world. Check for more info in my tip "Skibladner paddle steamer" in my "must see" tip.
The ship can be combined with a travel to/from Oslo to Lillehammer via the towns Eidsvoll, Gjovik, Hamar or Moelv in combination with bus or train. A popular outing is to travel by morning train to Hamar and take the ship back to Lillehammer and have dinner on the way.
There are buses that regularily run from Lillehammer Bus and Train Station to Nordseter, Sjusjoen, Gausdal and toward Dokka. All of these routes offer easy access to trailheads and major trekking and skiing areas, including some Grand Randonnee style trails.
The easiest way to get to Lillehammer is by train, for example, from Oslo. If you buy the ticket few days in advance you can save $€£. Check the official site for special offers.