As 4 me i can't put my child in a new private primary or secondary school sha no matter how attractive or beautiful the school looks.I need a school with experience and good track history;at least of about 10 yrs.
For primary schools i'll go with the following:
St Saviours School(Ikoyi or Ebute Metta)-Founded 1951
Corona School(Gbagada,Ikoyi,Apapa,VI)-Founded 1955
Grange School(Ikeja Gra)-Founded 1958
Grace Children School(Gbagada)-Founded 1968
Greensprings School(Anthony,Lekki)-Founded 1985
Then for secondary schools ;Atlantic hall,Olashore,Loyola,Adesoye,VF,Trinity,Whitesands,Lagoon,Grace High,Corona,Grange,Greensprings, etc
These are private schools with VERY high quality and have stood the test of time,with a good alumni record.
NB I hav to confess that the current school fees of most of these schools are VERY VERY high and range from a minimum of N500,000 per annum-Day fees to about N2m for boarding!!!!God will help me

Definitely Private Schools,ESPECIALLY for Primary education,infact there are differnt categories of private primary schools.
Then for secondary schools,i'm a bit more conservative and i stiick to my self made top 20 private secondary schools in Lagos/Nigeria.If my child can't get into those top 20 private schools,then i'll rather send my child to KC,QC,ST gregs,Holy Child,Air-force or ISL.

I forgot to put
12.)Vivian Fowler memorial College

After thorough research,these are the top private/independent seconsary schools in lagos:
1.) American International School
2.) British International School
3.) Atlantic Hall
4.)Corona Secondary School
5.)Dowen College
6.)Lagoon secondary school for girls
7.)Grace High School
8.)Grange School
9.)Greensprings School
10.)ISL
11.)Whitesands School for Boys