for me, the area you are looking for is called "Medical academy", in particular between MMA (Military Medical Academy) and Pirogov. It is like to be downtown, but not exactly in the center and you can still park for free It is 15 minutes on walk to NPC (National Palace of Culture) and Vitosha str., it has a lot of public transport and in a year or two, there will be 2 metro stations. There are schools, kinder gardens, many shops, cafes. The area is safe (for Bulgaria) and prices are OK for such location. Such apartment would cost 400-800 EUR depending on the property.