I just moved a month ago from Madrid to Atlanta. Madrid is a great city and you will have beautiful weather at that time of year. Sunny and warm (or hot!)
To go from Barajas (the main Madrid airport) you can take either the metro or a taxi, depending on the amount of luggage that you have. Going from the airport to Gran Via on the metro will be at least 2 subway changes. The metro from the airport is line #8 which you would take to Nuevos Ministerios #6 line to the #2 line which will get you to Gran Via.
That said, I recommend a taxi. There are always 100's of taxis lined up at the airport and most are honest (low ripoff potential). The meters run well and are relatively inexpensive. This ride will cost you around 20-25 Euros. No need to tip, just round up the bill to the next Euro, unless you have an obscene amount of luggage and they help you a bit more.
Gran Via is a great neighbourhood for a short visit to Madrid. It is within walking distance to the Prado, Reina Sofia and Thyssen museums. Also to Puerto del Sol, Plaza Mayor and the Palace. Before moving to MAD I used to stay at the Hotel Atlantico (Best Western) many times. Within your budget and clean and good location.
Things to do:
Palacio Real at Plaza Oriente is beautiful and still has lots of furniture and artowrk. Still used for state events.
Tapas - walk along C/ Cava Baja - behind the Plaza Mayor for lots of great little restaurants. Avoid the places right in the Plaza
Food -
Spanish: Taberna del Alabardero: Plaza Oriente. Great food - both meat and seafood, amazing jamon, large wine selection. Moderate price
Basque: Expensive
Goizeko Kabi c/ commandante zorita, 20
Fancy basque restaurant. amzing seafood and innovative desserts
Continental: Inexpensive
Bazaar - c/Libertad, 21 (behind the Chueca metro stop)
varied menu, fun atmosphere and cheap wuality food
The metro is a gret way to get around. Clean and safe. Buy a 10 trip ticket if you are there for more than 2 days and 2 people (you can share). This costs about 5,50E (price might have just gone up)
Let me know if you have any specific questions.