For the vast majority of students, the sightseeing in Madrid starts with its central part. We can find in the central part a large number of interesting places and buildings, like the Royal Palace or the wonderful Plaza de Espana (located west of town), the Paseo de Castellana (to the north), the Atocha Station and the Rastro flea market. Many football fans also visit the Estagio Santiago Bernabeu, located on the Paseo de Castellana.
The formal center of Madrid and Spain is the Puerta del Sol. The Puerta del Sol is sort of the local Times Square, and it is the first place from where Madrilenos measure highway distances.
A few blocks southwest we find the Plaza Mayor (following the arches that lead to Calle Mayor). The Plaza Mayor is an impressive pedestrian stone paved square, featuring covered passages on every side. The plaza project was initiated in 1590 and finished in 1619. It was then later rebuilt (1790) after a serious fire. (Fires were pretty common during Inquisition times).
If you head west following the Calle Mayor, you will find the Plaza de la Villa. Inside the Plaza de la Villa is an old City Hall (Casa de la Villa) built in 1630. These buildings can be visited Mondays at around 5 PM.
Walk farther west and you will find the Almudena Cathedral, an impressive building that was first introduced in 1879 and completed much later, in 1993. In the same area you will see the Royal Palace, built around 1750, after the Moorish alcazar was burned to the ground.
A complete tourist program is offered by the Madrid Tourist Board. Walking tours are in English, Spanish and even other languages. You will find reasonable prices, from only a couple of euros for a quality tour of about an hour and a half.
Private tours are organized by the Wellington Society, perfect for small groups and families. Beware, though, these prices are substantially higher than other tours.
Sightseeing busses are offered and operated by Madrid Vision. The program features two routes: the first one set around the old center, and the other around the Real Madrid soccer club, as well as some other parts of the new Madrid.