Hello
I live in the area of Sagrada familia, a part of te Eixample. Is very near to the center and the public transportation is good enough. There ara a lot of good schools and kindergarden here. About green areas, you have to know that Barcelona ins not a very good city in this sense, there are lot of small square but not very big green areas, with some expemptions, as Parc Güell and la Ciutadella.
Now is the time to apply for the kindergarden, but it depends on the age. Younger of 3 it will be very difficult to find somenthing in june, but if is older, public adminitration is obligated to give you a place. Maybe you can try to have a place by internet, if you want I can tell you some kindergarden in this area.
good luck