I just moved to Madrid from NYC about 4 months ago so I can help with the apartment hunt ideas.

The rental market in Madrid is difficult and getting more expensive everyday. Your budget will have an impact on your neighbourhood options as well as your needs.

The northern neighbourhoods tend to be nicer, safer, and have more green - but then they are also more expensive.

I saw several 2-3 bedrooms (small bedrooms) in the Arturo Soria area for 800-900€ a month. Many of these had doormen and a small garden or pool area for building residents. I now live in the Arturo Soria / Conde de Orgaz neighbourhood which is prolly one of the more expensive areas but very very safe. I wanted a pool, a terrace, and a parking space so I am paying more than in other areas.

I used a realtor and found it very helpful. You can also try walking in a neighbourhood you like and stopping whenever you see a sign offering a flat for rent. These signs are quite common and can save you the realtor fee.

email me if you more details