The city of Baku is located on the hot and dusty Absheron peninsula. The center of Baku is a patch of green in a largely brown area because of the many liters of water that is piped to the city every day. The city has a metro, a well developed bus and mini bus network, and a large area for walking including the Inner Walled City, The Boluvar, and Fountain Square.
To prepare for traveling in Baku City center; Comfortable shoes, shorts, t-shirt and some sun-cream wouldn't go a miss. Read some articles online to get a feel for the city and how to get about, taxis are very easy to find and most drivers even though they don't speak English still have a rough idea, taxis within the City are cheap. Make sure you have your emergency numbers on you also.
Check into lodging
Eat lunch at Fountain Square (eg. doner kebab from Lebanese fast food) and walk around the square
Walk towards Caspian Sea and walk along the boulevard
Visit the Park Bulvar shopping center, and Crystal Hall (Where they hosted Eurovision 2012)
Go to the Old City and climb the Maiden's tower, and spend the rest of the afternoon walking through the narrow streets and visiting the Palace and old mosques.
Have dinner in the Old City (Karvan Saray for Azeri food in a unique ambience, Sultan Inn rooftop or Meridien hotel rooftop restaurants for european/fusion with excellent views) or head back to the Fountain Square for greater variety of inexpensive cuisine.
Go to the Jazz Cafe (live jazz at 9pm), a movie at the movie theater in Fountain Square (in Azeri mostly), or club/pub/bar for a drink, or two.
Take metro/cab to 20 Yanvar metro station. At metro exit, turn right and right around the block to get to Chudo Pechka. There's plenty of sweet/savory pastries to choose from (take out only). Walk across the road to the park with the 26 Commissars monument and enjoy the food there.
Alternatively, at Sahil metro exit, turn left and walk one block to Cafe Aroma for coffee/tea.
Go to the Ateshgah Fire Temple via electric train (leaves 28 May station at 10:20) Alternatively, marshrutkas depart from Qara Qarayev or azizbeyov metro stations
Return to 28 May station and eat a lunch of Georgian food at Georgian Home
Go see mud volcanoes and petrogyphs in Gobustan. Gobustan is about a 45 minute one-way taxi drive from Baku.
Eat dinner at Cafe Rendezvous
In the evening, go to the opera at the national opera on Nizami street. Otherwise, go to the rooftop bar at Sultan Inn in the Old City for a drink and excellent views of the Maiden Tower, Caspian sea and the TV tower.
Check out of the lodging
Buy pastries at Chudo Pechka by for breakfast and eat them in the nearby park
Go to 20 January station and catch a mini-bus to the next destination