We've travelled to Seville on and off for the past 13 years and usually stay 6 weeks or more. We tend to visit between November and March. We combine Seville with other cities/towns.
We've stayed in most neighborhoods. All have been excellent for various reasons. It depends how far you want to walk.
We choose our accommodation nowadays based more on the actual street as opposed to the neighborhood. We know the quieter, quainter central streets. These can be a 5 min walk from main thoroughfares that are packed daily with tourists and locals, but you would never know you were amidst the chaos of Seville central.
Bario Santa Cruz, Alfalfa, Arenal, La Alameda, Magdalena, all are excellent neighborhoods located an easy 5 to 15 min walk to most sites. All close to hundreds of tapas bars, restaurants etc. Supermarkets are everywhere. Mas, Cortes Ingles, Carrefours, the usual suspects.
If I had to pin it down I would probably choose El Arenal as one of our preferred neighborhoods. But it depends what street you stay in.
As an aside, cycling along the Guadaquivir river is truly spectacular, if that's your thing.
I can suggest some accommodation options with kitchens if this is of interest.