Oslo Airport (OSL) is the closest airport. There are many direct flights to different destinations in Europe and overseas. SAS (www.sas.no/en) and Norwegian (www.norwegian.no) are the main air companies. From Oslo Airport, take the bus or train to Elverum.
By bus (the fastest and cheapest way of going to Elverum):
From Oslo Airport, take 'Trysil-ekspressen' directly to Elverum Rutebilstasjon (the main bus station). It takes about 1,5 hours . Here you can find the time table for "Trysil-ekspressen".
From Oslo Bussterminal (the main bus station), take "Elverums-ekspressen" directly to Elverum Rutebilstasjon. It takes about 2 hours. Find the time table on www.elverumekspressen.no
From Oslo or Oslo Airport, take the train in direction Hamar-Lillehammer. Change train at Hamar and take 'Rørosbanen' to Elverum. Oslo Airport - Hamar: 1 hour. Hamar-Elverum: 30 minutes. You may have to wait for some time at Hamar.
You can find time tables, connections and prices on www.nsb.no.
From Oslo, follow E6 in north direction. Take on R3 before you get to Hamar (direction Elverum). Around 2 hours from Oslo.