I started gathering some information about Iceland reading Travel.SE (what else?) and I was quite surprised I didn't find a single question about moving around Iceland using public transport, a car seems mandatory.
How widespread is the public transportation system net in July? Would it be viable to rely upon it to move around the Ring road and the Golden circle? Is there a bus (or a train?) which goes in the central area of Iceland?
You are correct that most people travel by car. The main roads are excellent and signage is impeccable. The best I've seen anywhere! There are even maps along the road every few hundred kilometers.
The other main form of transportation are buses. Within a city, they are easy to take and there is also an extensive network of bus routes. Most, if not all, are available from June to August. The central station in Reykjavik is very well connect but as you go to most other cities, you basically encounter buses going in 2 or 3 directions at most. It's hard to get lost!
The main issue if with timing, inter-city buses have fixed schedules which are not that frequent. I recall 1 to 4 times daily, depending on the popularity of the route. The tourist-oriented options do a circuit which lets you see more than one site in a day without having to time the buses yourself.
When you consider timing, your schedule not only gets dictated by bus timetables but you also arrive and leave exactly at the same time as everyone else. You lose the sense desolation and immersion with nature is lessened as the hordes of people coming out of buses are large and noisy.
You can bridge some gaps in the bus network with domestic flights and even ferries. Usually short-cut between fjords. These are certainly not cheap and neither are car rentals by the way. I met a few people who brought their own car from Denmark since there is a thrice-weekly ferry which lands on the east side of Iceland. One of them basically drove from Italy in order to have is own car there.
There are no trains; Iceland has no rail-based public transport.
The systems that do exist (bus) are under-developed, and while there are nation-wide bus and coach services, many of the locals will use internal flights to get across the country.
Iceland actually has one of the highest rates of car ownership in the world (similar to the US) which has led to little need for a public transport network.
Both rail and light rail systems have been proposed, but as yet are not being constructed or in service.
There are some ferries between port cities, as one alternative to get around.
Primary info source: Public transport in Iceland
Saying that, there are coach/bus companies which serve the various towns, and you can use them to get around, and the bus network is reasonably extensive.
Some coach companies:
the bus network, and timetables are available on Nat.IS.
