Mainland South Iceland itself is not served by any flights, domestic or international. On the western side, it's within reach of Reykjavík, which has a domestic airport that connects with some of the towns in different parts of the country. The eastern half of South Iceland can be reached by flying from Reykjavík to Höfn a town in the southeast with Eagle Air (http://www.eagleair.is). This is close to Vatnajökull National Park, Jökulsárlón and other popular destinations.
Vestmannaeyjar have a domestic airport, with flights from Reykjavík with Eagle Air.
Taking the ring road east from Reykjavík will very quickly lead you into South Iceland. This is how by far most locals will choose to travel and as with the rest of Iceland, having a car of your own will give you a lot more options for exploring the region than relying on public transport. When entering from Reykjavík, you also have the alternative of taking the road through Þingvellir to enter South Iceland. Drive up on the ring road through Mosfellsbær a town on the outskirts of Reykjavík, as if you were heading for West Iceland, and following the signs indicating Þingvellir road nr. 36.
If you're entering the country by ferry and bringing your own car, you will drive off the ferry in Seyðisfjörður in East Iceland. Once you get to the ring road, head southwards and you should be able to reach South Iceland in a few hours. It may be best, though, to stop somewhere around Höfn in the southeast before heading further, as the distances are big.
The Reykjavík area bus system Strætó (http://www.straeto.is/) reaches all the way to Selfoss. Take bus nr. 51 from Mjódd, which goes over Hellisheiði to the town of Hveragerði and then terminates in Selfoss.