I would recommend that you stay on the city center, or el Centro, which encompasses Sol, Opera,Plaza Mayor, Huertas, Gran Via and Chueca areas. This area is filled with tapa bars and wonderful inexpensive restaurants. The museums, the Plaza Mayor and the royal Palace are within walking distance of each other. But taxis, buses and the metro are available and affordable.
A very nice hotel, where I stayed in my last visit 2000/2001 was the
Hotel Suecia. it is in a very safe, and elegant area and very close to the Museo del Prado, the Thysen, and the Reina Sofia museums. It is also within a few blocks of Kilometro Zero , a.k.a. the Puerta del Sol. I was able to pay very little for it because I participated in a Hotel Discount Scheme called
Bancotel . I believe it goes for 50 euros per night. If you go to the website you will see the participating Hotels. However, there are also wonderful Hostales available. Hostales vary in prices, but are generally more reasonable than hotels.
If you use the
search function above, you will find many threads with excellent advice on accomodations, restaurants and things to do in Madrid, from members who recently traveled to Madrid.
June is a splendid time to visit Madrid. Hope you have a wonderful stay in Madrid.