Many trails start from Break Sokos Hotel Koli (booking.com). It is located on a cliff and we really wanted to spend the night there, but there were no free rooms. Therefore, we drove to the parking lot, took the funicular to the hotel, and walked along the trails.
Check the official website, for a detailed description of the trails and other relevant information.
Getting there: parking N 63 05.685 E 29 48.184. There is no public transport, but the nearest stops can be seen here https://www.nationalparks.fi/en/kolinp/directionsandmaps.