Depends what your plans are....this would decide where best to stay.
the weather's not going to be extremely hot that time of year so i doubt that you'd like to be near
find an apartment near Les Rambla....the main road in Barcelona. many things are walking distance from here...shops, eating, architecture.
if you want somewhere more quieter then an area called Diagonal Mar or olympic village: nice area, some restaurants and supermarket to do your shopping. there's a metro station near which is useful but it's in a quieter area. you will have to take metro everywhere.
if you're vegetarian....be prepared to eat lots of chips, veggie paella and salad and i would advise to learn the spanish for "vegetarian" "no egg" "no fish"