My husband and I have decided to relocate from San Francisco to Madrid as soon as we have things tied up here. If all goes as planned that should be possible by the end of 2021/beginning of 2022.
Even though it's still a ways off in the future we've started looking into different neighborhoods where we'd like to settle. We already know we love Chamberi and sort of have our hearts set on Trafalgar, but are investigating the rest of the neighborhood as well. We've also done some preliminary searches into Arguelles, Palacio, Embajadores, Conde Duque, Jeronimos, Castellana, and Recoletos. We'd like to be near the city center but not right in the thick of everything. We're both in our 40s and coming around to 50 faster than we care to admit so, while we want to have easy access to the center, we are looking for somewhere a little bit quieter.
If anybody has some insight about what daily life is like in any of these neighborhoods, or thinks we may be missing out on a different, great neighborhood we are all ears.
Trying to pick a neighborhood
Maybe check out La Moraleja, just outside Madrid. Lots of expats live there because of international/bilingual schools. Lots of gardens, semi-gated communities, chalets. Not cheap though and honestly a bit posh (like the Salamanca neighbourhood in the city centre).
If you like trendy neighbourhoods Malasaña and Chueca should be on your list.
But honestly if you like quiet you should avoid the city centre all together because the Spanish come alive at night and they have a habit of talking quite loudly 
.
Good luck with the search!
The OP didn't mention kids / school so not assuming anything.
Before settling down I stayed in different areas for 4-5days each and got a feel of everywhere. From the age range and your preference for close to the city but not in the thick of it, Chamberi would be the top choice. In fact I also liked Trafalgar the best when I saw the big 24/7 Carrefour! I took a look of Salamanca but I found the residents too old for me. Malasaña and Chueca are too touristy and young.
Recoletos is beautiful and green and next to Parque de Ritero as you know but it's too quiet for me personally. More restaurants bars and supermarkets in Chamberi..Streets in Palacio are really beautiful with old architectures and historical streets but the layouts of the old buildings are just not practical with many windowless rooms. Embejadores is the counterpart of Chambri on the south side of Centre but just a tiny tad lower scale. Still pleasant and convenient and very accessible to the Riverside park along Madrid Rio.
Chamberí is a great neighborhood. I would also recommend Salamanca. It's a quiet neighborhood close to Retiro, one of the most beautiful parks in the city
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