Triana is considered the soul of the city, it is a blooming neighbourhood just next to the centre. If you want to live as the locals do, this is your choice.
Nervión is another area considered as the modern centre. This neighbourhood is quieter than the centre itself and Triana, but it has all the things that you want (shopping malls, restaurants, supermarkets, schools, etc.). It is also the district of Sevilla with more green areas.
Los Remedios, also a modern district as Nervión, it is a good choice if you are looking for a quiet place where to live (the busy centre and the tourists visiting the city are far away).
La Alameda de Hércules is the art district, with local shops in its narrow streets. This is the place where you can find 100% Andalusian houses.