The Puerta Del Sol and Plaza Mayor are very close together and in the tourist heart of the city. As plazas go, the Puerta Del Sol is like many others. The difference is it is great for people watching. It is also the primary location where groups get together if wanting to protest or have their voices heard. It, and the Plaza Mayor, can get very crowded so there are plenty of pickpockets operating in the area. It is very much worth visiting, but more interesting if something is going on.
Also, the area between the Puerta del Sol and the Callao metro station (along Calle de Preciados and Pl. del Callao) is a main shopping area. For a good view of the city visit the rooftop 9th floor of El Corte inglés, gourmet experience. It’s at Pl. del Callao, 2. There are bars and restaurants if you want that, but is nice to visit for the view. Some shady characters and sex workers are frequently noted in that area.
There’s little of interest between the Reina Sofía and Prado museums along Paseo de Prado/Recoletos. Continuing past the Prado, if coming from the Reina Sofía, are the Alcalá gate (a block off the Paseo) Fuente de Neptuno, and the Plaza de Cibeles. A little further at Plaza Colón is the Rana de la Fortuna sculpture, and the Fernando Botero sculpture, Lady with Mirror. There is another of his sculptures just outside terminal 1 at the airport.
A few blocks up from the Prado is Retiro Park. It too is a good place to people watch and get away from the hustle bustle of the city.
When up by the Royal Palace you could make the walk over the the Temple of Debod, then the Plaza de España (statues of Don Quixote and Sancho Panza) before heading back towards the Gran Vía.
There is no order you should see those places, but I wouldn’t visit both art museums on the same day.
