Hi, 3 adults (parents and adult child) traveling in October primarily to Valencia but we will be in Madrid the first 2 nights. Looking for a neighborhood or area in which to find a hotel that we can easily enough get to from the international airport, then to the train station a couple days later, and where we can enjoy walking (or some public transit) around the city the days we're there. So someplace fun to be but also easy to get to / from plane and train. Thank you!!
We have always used Hotel Europa. About as convenient as they come. A hundred yards from the subway and easy access to everything. Just north of Puerta Del Sol and a couple blocks from Plaza Mayor. It sits are the end of a pedestrian zone. With three adults, the taxi probably is cheaper than the metro. Madrid taxis are fine.
We also had a nice stay at Hotel Europa
Madrid has a really good metro and commuter rail system so I wouldn't worry too much about proximity to the airport or Chamartin (train station for trains to Valencia). I also recommend the ride-hailing app "Cabify" for Madrid and Valencia if you need a taxi.
As far as neighborhoods, I really like Barrio de las letras and Chueca. I like the Only You boutique hotel in Chueca, and the Petit Palace chain (their breakfast is pretty bad if that's an issue). Also, the Radisson RED Madrid is pretty good.
I can recommend the H10 Villa de la Reina hotel in Madrid. Large breakfast, free cava in the evening and little candies and tidbits in the lobby during the day. Rooms are wildly different. Staff absolutely kind and helpful. We also really liked the Circulo Gran Via. Our room was massive with an enormous king bed and the bathroom was also large with wonderful products. Both on Gran Via right in center of everything.