El Putxet is right is near Sant Gervasi La Bonanova, which means you are still in a posh silent clean area of Barcelona where you will not be near the Metro.  Most likely,  you will need a car.  The pluses of the Zona Alta (the term used to refer to Sant Gervasi, Sarrià) are: (a) silence, (b) silence, (c) silence (d) cleanliness (e) extreme cleanliness (f) silence (g) people have domestic help ... and yes, there are good private schools there.  But it is possible to live elsewhere in the City and send your child to a good private school, whether in Sarrià or elsewhere.
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I’ve been living in the Sagrada Familia area and its perfect in my opinion. A lot less noisy and hectic than Las Ramblas but still very central and with easy access to the beach. Its not the cheapest area but |I love living in this area.

Exiample is a very good alternative and certainly worth considering.

I would also recommend the Sagrada Familia area if you can. Its a popular area to live in so it may be quite hard to find a suitable place but I would definitely advise allowing yourself plenty of time to find somewhere as it can be pretty competitive over there!
I would recommend casamundo.co.uk if you’re looking for an apartment to tie you over. They have quite a few accommodation options available in Barcelona so may be worth having a look about.

I’ve been in Barcelona for 10 days and it’s really wonderful. Wanna come back soon.

I recommend Sagrada Familia area too. I’ve friends living there and it’s nice to live. Sants is a good option if you can afford and need to travel around Spain/other countries because the train and bus station are there.

Is there are anyone here lining in Sant Cugat Del Valles. Want to learn more about this town.