searching out family roots near Stockholm

Hello
The present-day municipalities of the area are Nyköping, Trosa, Gnesta, and Södertälje (and Oxelösund). A bit inland are Katrineholm and Flen Municipalities.
Nyköping is a provincial town of around 30,000 people, with a county museum, a castle (tower, ruins), some surrounding islands and shore areas of recreational interest (notably Stendörren nature reserve). The Swedes tend to associate the town with things like Nyköping liqour brand (flavoured vodka), Nyköping Banquet (a historical event), Studsvik nuclear power research facility, the Skavsta airport (with one of Ryanair's hubs, plus Wizzair etc, plenty of destinations), and (at least earlier, as a travel destination popular with Stockholmers) numerous lakes a bit inland, including or near Lastringe and Vardinge areas. It seems Mölnbo is the largest settlement in Vårdinge socken (parish) but also part of Gnesta.
http://en.wikipedia.org/wiki/M%C3%B6lnbo
http://en.wikipedia.org/wiki/Gnesta
http://en.wikipedia.org/wiki/S%C3%B6dermanland
Järna isn't far away either
http://en.wikipedia.org/wiki/J%C3%A4rna
In Lästringe socken there's a village of the same name which has (according to Swedish wikipedia) about 50 houses and nearby is a 17th c mansion named Oppeby.

Trosa is a small town known for particularly its smallness. It's worth a visit as it is considered well-preserved but I'm not sure if there's a lot of sights there. Other, nearby towns of similar qualities are Mariefred (with Gripsholm Castle), Strängnäs, Arboga, Söderköping, Vadstena, Skänninge. The Göta Canal is a bit south of Nyköping and can be visited north of Linköping at Berg for instance.

Södertälje isn't all that interesting but has an outdoor museum. Norrköping is overall more atmospheric. Kolmården is a leading zoo about 40 km south of Nyköping in a forested area (20 km northeast of Norrköping). A nice drive of the area can be taking the (free) car ferry from Jönåker to Vikbolandet, then again to Stegeborg (ruin) and back via roads E22 and E4.