Bílar og fólk Bus Line 2 runs twice daily June 1 - August 31 from Reykjavík bus station via Selfoss. Reykjavik Excursions also runs seasonal (June 15 - August 31) passenger (i.e., non-"Golden Circle" tour) service from Reykjavík via Þingvellir. Finally, roughly from June to August, Bílar og fólk also runs overland buses between Reykjavík and Akureyri that stop at Geysir and Gullfoss. From Reykjavík, figure on a fare of roughly Kr 3000-3500 per person 1-way.
Beware: Bus schedules change quite frequently, and the websites are not always updated in a timely manner. Never count on any one bus to get you to or from your destination exactly when the websites say you will. Always allow slack time for any time-sensitive activity (such as a flight). If all else fails, consider renting a car from the Budget rent-a-car office across the street from the bus station.
Most visitors who come here arrive via tour buses like this one. I wanted to try to see the sights of the Golden Circle Tour without actually joining a tour group, but when I calculated the costs of renting a car and doing it on my own, it really wasn't worth it. Unless you're traveling in a group or five or six people, renting a car isn't going to be cost effective. Of course, if you want to take the Rim Road around the entire country or if you just want to do things on your own schedule and money is no object, then by all means, rent a car!
I used Iceland Excursions to do the Golden Circle Tour (6600 Kr, 8 hours in duration) and had a very good experience with a knowledgeable tour guide and a comfortable bus.