Enhorabuena! What an exciting summer you have coming up!
Salamanca is a very nice neighborhood, but not as green as some others. With a new baby you'll want to take daily walks to the park, along with the other new parents. It's a great way to meet people. I would suggest you get a map of Madrid and look for any green areas north, northeast, and northwest of the center. Try to find a place close to a park. Some areas to check would be Retiro, Parque Oeste, the Arturo Soria area (same metro line as Serrano), the areas around metros Pio XII, Colombia, Concha Espina. Farther north - try the newer areas of Barrio del Pilar, or even Mirasierra, which has beautiful apartment buildings in the back, with clear views of the mountains, and little "city" feel.
As far as the prices of rents go, I'd check Segundamano, either on the web (
www.segundamano.es ), or, if you can get a Madrid issue in Barcelona, on paper. The website is nice because you can see photos of the pisos. Unfortunately, the closer you get to a park, the higher the prices go! I'm afraid I've recommended some of the most expensive areas of Madrid. But even if you're 5 or 10 minutes away from a park, it's worth it.
I saw a forum just yesterday at
www.expataccess.com , checking a link someone here had recommended. It may be just what you're looking for.
Good luck!