In South Belfast (Stranmillis, Malone, Lisburn Rd) you have Stranmillis, St Bride's, Finaghy, Cranmore Integrated, Holy Rosary, Botanic, Fane St, Taughmonah. The first two in particular are, ime, excellent schools, although are quite oversubscribed.
Also in that area, fee paying schools include Fullerton (Methody feeder), Victoria (girls school, feeder for Victoria College) and Inchmarlo (boys school, feeder for Instagram). I know people with kids at Victoria who also rate it but I'm not personally familiar with any of them.
In terms of feeling family friendly, good chunks of Stranmillis are always well loved by families, the Malone road area is great and near parks etc but more expensive. There are new developments further on out the upper malone road (st Anne's primary out that way is excellent too), and as a pp said, four winds, carryduff has lots of family estates there.
I'd second checking the eti inspection reports although some of the more recent ones are bare bones as there was a teachers strike on during them! Also checking with schools if they're likely to have places coming up. My kids' school is near QUB and the City hospital so we often have families of visiting lecturers/doctors here for a couple of years then going so you do get places coming up each year.
Hth, feel free to pm if you've any questions.