Every time a friend comes to visit, I tend to give out the same travel information. Being lazy, turning that into a posting seems sensible.
While I no longer live in Helsinki, there is no real need to delete this post. Keeping what the migration tool spit out.
This page shows you all possible ways to get from the airport to the centre of town. My preferred in the Finnair City bus, you can find it’s schedule in plain text here. A taxi ride from the airport to my home costs me between 35€ and 40€, while the bus costs under 10€. Both take pretty much the same time for the journey.
If you fly to Tampere, you will need to take public transport from the airport to Tampere itself, and from there on a train to Helsinki. That train journey is 1:26 or 1:46 hours, depending on which connection you take. Schedules for Finnish trains are found here.
If you fly to Tallinn, then you’ll have to go from Tallinn to Helsinki by boat. You can find schedules here.
For the adventurous or impatient, you can also take a helicopter.