Belgrano is probably your safest bet, I wouldn't put my hands on the fire for any neighborhood but Ive lived in Belgrano and it's one of the nicest neighborhoods.
One of the best schools you could take your kids to is St. Catherine's School, I don't know how much it is per month there anymore but it's one of the highest rated schools. I went to that school, they have like 3 different schools in 3 different areas of Buenos Aires and they have one of their schools in the UK. They've got a club, a pool, polo, hockey, tennis, and the study level is just the same as in the states, which will be quite hard for you to find here in Argentina as the pacing is different as far as education goes. It also has a choir, an outstanding building, a very good cafeteria, and they teach Spanish, English, French, Italian and German.
The only downside is that it's a catholic school so if you ain't catholic you might find this not suitable for your kids, if you don't care about this at all or you are catholic then I'd highly suggest this school, you wouldn't want your kids grades to drop when you get back into your homeland.
Statal schools aren't very good unless it's the Lenguas Vivas, which they will need to first pass an exam to join. Although they will need to pass an exam to join St. Catherine's as well, Lenguas Vivas is the best state school. I wouldn't recommend the other's, the English level is poor and if they know no Spanish they will be subject to being bullied and having a really bad school year.
People here are very accepting but kids will forever be kids and you want your kids to have a good year, so I'd highly recommend against state schools unless it's the Lenguas Vivas in which they speak English fluently and have actually quite an amount of people from other countries studying abroad (they even got an exchange program).
These are my 2 cents, I hope it helps.
Btw, if you'd like to know which parts of belgrano are the nicest lemme tell you the area around the train station ''Belgrano R'' is the nicest and most residential-quiet yet comfortable area you could live in. It's really chilled, it has a beautiful park called Plaza Castelli, tons of restaurants and even though it's so suburb-ish like it's 30 mins from downtown, nearby about everything.
=) cheers.