It looks like the tickets to Skavsta are a bit cheaper than those to Arlanda.
How do I go to Stockholm from the Skavsta Airport?
The easiest and most straightforward option would be to take the airport coaches (Flygbussarna).
They leave from a couple of different places in Stockholm and seem to take 80 minutes to get from the center of Stockholm to Skavsta.
It's 139 Kronor for an adult one way, and 278 for a return ticket.
Transit options from Stockholm to Skavsta Airport are listed in some detail on the airport's website. I am assuming you do not have your own car, in which case your options seem to be to take a taxi from Stockholm directly to the airport, or to take a train from Stockholm to Nyköping, and then either local bus 515 or a taxi from Nyköping to the airport.
The direct airport coach costs SEK 139 (SEK 119 if under 18) if bought online.
Alternatively, you can take local bus 515 to Nyköping and then a regional train to Stockholm. This may well work out cheaper if bought well in advance. Through tickets from Skavsta are sold by the national rail operator, with discounts available for 20-25 and 16-19 year olds.
A ticket from Skavsta to Flemingsberg, 15 minutes before central Stockholm, is even cheaper: your train ticket is checked between Nyköping and Vagnhärad, and Flemingsberg is a good while after that, so you'll hardly get caught.