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.