Hi, Marcie--
I have stayed in two hostales (one was Pensión Lisboa, and the other MIGHT have been Pensión Las Vegas, but I am not sure!) on Calle Melendez, which is just one street off of the Rua Mayor (which runs south from the Plaza Mayor). Both were inexpensive and fine--but certainly nothing fancy! They had (if I recall correctly) private bedrooms and shared bathrooms. The street itself is small and short, running down toward the Clerecia. These places might be a step up from a youth hostel with shared rooms.
You should check out the thread
Lodging report: Salamanca for additional suggestions.
I have also stayed in Hotel Las Torres, right IN the Plaza, and wouldn't recommend it unless you have a strong desire (as I did) to be looking out over the Plaza. It's my favorite place in the world, so I thought it would be fun to spring for a more expensive place in order to overlook the Plaza. However, the room itself was only so-so, and the bathroom was tiny and quite worn. I think some rooms there have been remodeled, though.
Anyway, if I were going back (I've been 4 times) I think I would choose an inexpensive hostal and save my money for a shopping excursion or nice meals.