I'm planning to rent a car for a road trip in Spain for this coming August, but I could also consider just going from town to town using bus, if available.
Is it possible to get from any city to any other city in Spain by public bus? How about from small towns to larger, nearby cities? Or would a car be needed? Is there a centralized website where I can see fare and time information?
Spain has a very extensive intercity bus system, in fact they have one of the larger bus systems in Europe. You'll likely find it easy to travel without a car.
That said, it depends where you go. There are very different levels of service across regions and specific cities. If you're close to a beach or a city you'll likely find plenty of bus connections, but if you're in a more rural inland area the options can be limited, to a single bus a day, and some places just aren't covered by the bus network.
Spain regulates their domestic bus system meaning, with a few exceptions, only one operator is allowed to serve any given city pair. This makes longer journeys more complicated, because the point where you need to make an interconnection between two lines of different companies is often unclear. The regulatory system differentiates between longer routes, connecting various autonomous communities and shorter routes, within a single autonomous community. Each route is given out from the government to private bus companies as a contract for a certain number of years, prices and level of service are part of the agreement, the great thing about that is that the network doesn't change often and service is very reliable.
In contrast, on international routes to and from Spain you'll find competing operators who change their schedules quite frequently, but competition means they often have more aggressive pricing and better amenities.
At Busbud, we're building a centralized booking system for buses in Spain. The routes, schedules and tickets for the largest bus companies (ALSA, Avanza, and many others) are up on our site and apps already, and we're adding new companies every week at this point. We've identified over 200 companies that operate regular routes, we're working hard to get those all viewable at once.
Buses between cities are easy and generally cheap to do, and there are many buses adapting basically any timeshift you want... however, excepting a few major cities, public transport for moving within the cities (specially if you go to the suburbs or surrounding towns) is generally an issue: taxi cabs are typically very expensive on those cities (it's cheaper on big cities with nicer public transport -metro, bus, tram-, but on those with rather bad public transport, they have pretty high fares), and buses can be very delayed and kind of expensive for isolated tickets.
If you go to rural areas, forget public transport: it might range from bad to unexistant (and that's why basically every family in Spain except on those big cities I mentioned has at least one or two cars).
There's no good answer for your question unless you are more specific: Spain is pretty big and there are many differences between visiting cities like Seville, Valencia, Madrid or Barcelona, or visiting more rural areas or smaller region capitals.
