Public transportation in Stockholm is generally well functioning and safe. All public transport travel needs to be prepaid and you can buy tickets for public transport at the central station, where your bus or train from the airport arrives. Find information about the different kinds of tickets available here: http://sl.se/en/fares–tickets/
There are buses and trains that take you to Stockholm central station from Arlanda Airport. The bus takes approximately 40-45 minutes and costs between 99 (prepaid online) and 119 SEK for a single journey. Tickets can be bought online via https://flygbussarna.se/en or at the airport. The Arlanda Express train takes 20 minutes and costs 280 SEK for a single journey. Tickets can be bought online in advance via www.arlandaexpress.com / or at the airport. Most taxi companies offer a fixed price form Arlanda to central Stockholm, somewhere between 450-550 SEK, but check with the driver.