Seasonal activities

All year activities.
Sight seeing: places like Seljalandsfoss, Gljúfrabúi, Skógarfoss, Reynisfjara, Dyrhólaey, Skaftafell national park and Jökulsárslón if you are going east and Gullfoss, Geysir, Þingvellir (Golden circle) if you are going west.
Glacier hiking: Hiking in Sólheimajökull with Arcanum.
Snow mobile tours: Snow mobile in Mýrdalsjökull with Arcanum.
Super Jeep tous: All year round you can go almost anywhere on a Super Jeep with South Iceland Adventure.
Snorkeling or scuba diving in Silfra with Dive.is

Summer activities.
Sight seeing in the highland Places like Þórsmörk (Thorsmork), Landmannalaugar, Emstrur and many more are accessible by private cars from the middle of June until September.
Hiking: Many hiking trails are in the south of Iceland. The most popular is Fimmvörðurháls a walk from Skógarfoss to Þórsmörk (or Þórsmörk to Skógarfoss), Another hiking is Laugavegur from Landmannalaugar to Þórsmörk (normally done in few days) and many more.

Winter activities.
Ice Caving in Vatnajökull with extream iceland.
Dog sledding with Dog sledding they also have summer tours on the glacier.