It's cliched, but definitely worth doing the key points on the 'Golden Circle' - Geysir, Þingvellir (pronounced with a 'th'), Gulfoss.
Then I'd head east out to Vatnajokull National Park, which is a big ice cap by the coast. The southern part is the most accessible. There's Jokulsarlon (iceberg filled lake, you can kayak in it also) and the Diamond Beach, beautiful walks (Svartifoss, a waterfall through big hexagonal baslt columns, is amazing), ice caves and glacier walks (get a tour, Skaftafell is good). Selsjalandfoss en route is a beautiful waterfall you can walk behind.
Blue Lagoon by the airport is a classic geothermal pool, but very expensive and touristy - still worth it though! (Remember to slather your hair in conditioner before you get in, the minerals in the water ruin your hair). I've not been, as it opened since my last visit, but Sky Lagoon in Reykjavik is supposed to be amazing - I'd probably chose that over Blue Lagoon. For a budget option in Reykjavik, their public swimming pool is geothermally heated, and has a series of hot tubs on a balcony outside. For a more authentic and natural experience, Secret Lagoon is fab, or there are hikes which go encompass hot springs (do your research though, and ask locals, because some can be very very very hot and are dangerous - don't just plunge into any steaming river you see!)