I plan to be in Madrid 10/3 - 10/11. First time visiting. We plan on renting an apartment, any neighborhoods that you especially enjoyed?
It's great that you have three months to spend there!
We stayed in Plaza Santa Ana in an apartment and found it to be a great location. Walkable to everywhere and great restaurants just outside your door.
We stayed in La Latina. We walked to Puerto del Sol. Enjoyed the neighborhood vibe and local cafes and bakeries. Also close to El Rastro.