Connecting between flights at Helsinki Airport is extremely easy, as all flights leave from the same terminal. Therefore all you need to do is follow signs for connecting flights, go through security and/or passport control (as applicable, see below), and walk to your next gate. You can remain airside in all scenarios (unless travelling on separate tickets and needing to re-check baggage).
Do I need to go through passport control?
Only if you are entering or exiting the Schengen zone (western and central Europe, excluding UK & Ireland).
A Schengen to Schengen connection does not require passport control (e.g. flying from Athens, connecting to Copenhagen).
A non-Schengen to non-Schengen connection also does not require passport control, as you never enter Finland (e.g flying from London, connecting to Tokyo).
Do I need to go through security?
It depends on where you have arrived from.
Passengers on flights from the EU, UK, US, Canada and Singapore do not need to go through security, and you depart the flight straight into the departure lounge – the walk to the next gate may be less than 5 minutes.
Flights from other countries do need to pass through transit security before being allowed back into the departure lounge, but this is efficient and should only take 5-10 minutes.