I just moved to Madrid about 3 months ago and it is a great city.
Depending on your budget, you can find most anything here in MAD.
Before moving here, I used to stay at the Westin Palace on business trips. It is at the upper end of the price range but very centrally located near the Prado, Reina Sofia, Plaza Mayor, etc. are all within walking distance.
For trains the AVE is the super-fast train. You can reach Sevilla in about 2 1-2 hours. Trains tend to be pretty good around the country and if you are staying just in MAD I highly recommend against renting a car. Traffic is a nightmare.
You should try to also visit Segovia and Toledo. Everyone raves about them.
As time gets closer feel free to email me with specific questions and I will try to find the answer.
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I Love New York -- Viva Espaņa