Nynäshamn is the furthest out zone (C) so you will need a single ticket costing 60kr for cash or 52kr from a vending machine (under 20s pay 36kr/32kr). If you can get the pre-paid strip of tickets costing 160kr then the same journey will cost 40kr. More than one person can share a strip of tickets and you can also use these on other forms of transport. Not sure if a bike costs anything. Journey time is just over an hour.

More ticket info in English:

http://sl.se/templates/PriceList.aspx?id=5302

Blackeberg is absolutely right, but here is a little extra info. The Nynashamn line is part of the pendeltåg (commuter trains) routes. On these route you can take a bike for free, however, you cannot travel with a bike between 6 and 9 am or 3 and 6 pm. The exception is during summer, where there is no travel restriction with bikes.

Bikes cannot get out or on the trains at T-centralen (central station), but from Stockholms söder or Karlsberg stations, which are stops either side of T-centralen on the routes, it is just a 10 minute bike ride.