Sol is crap for bars really, Plaza Mayor is a lovely place but the bars are a rip off and the food poor and twice the price of elsewhere. That's if you speak Spanish, if you don't you may well find extras or strange surcharges appearing on your bills.
Plaza Santa Ana is near Sol and a lot nicer in terms of having a drink. The streets around it, especially Calle Huertas, are packed with bars that are far less tourist-orientated. Alternatively, La Latina is very close to Plaza Mayor with loads of great tapas bars where most people around you will be Madrileños (especially two streets, Cava Baja and Cava Alta)