Star Alliance has four direct flights per day from Stockholm Arlanda (ARN) to the Tampere-Pirkkala International Airport (TMP). Bus number 1A takes you from TMP to Tampere city centre in about half an hour at about 5 Euros. Or you can just take a taxi to your hotel for about 25 Euros.
AirBaltic starts connecting flights to TMP via their hub in Riga (RIX) in April 2017. AirBaltic is not part of any alliance, but they have some code shares with selected Star Alliance members.
Flying via Helsinki-Vantaa International Airport (HEL) to TMP is also possible. There are four flights a day and these One World flights are code shared with practically all airlines in all alliances. Do not be tempted to take a taxi from HEL to Tampere, because it will cost several hundred Euros.
However, in some cases it might be more convenient to just take the train for that leg of the journey. The train ride should take a little over two hours and cost under 30 Euros. The trains run every day at least once per hour between 06:00 and 22:00.
The P train on the new ring rail takes you from HEL to the Tikkurila Railway Station every 10-20 minutes where you need to transfer to an InterCity or Pendolino train to Tampere. You can buy the ticket for the whole trip from a vending machine at the airport railway stop or from the train conductors. The buying the ticket from the conductor is a bit more expensive.