I am answering this only because I want the same information in another city and greatschools.com has not been a great website to find the real information on but everyone gives it anyway.
I am not sure what you are looking for elementary, middle or high. Public VS private is up to you.
Just so you know upfront my children go to private school, I am not wealthy I just want them to have a great education so that was my choice. I did a lot of research and picked the best choice for my family. Your situation and needs could be different.
There are some good private schools. There are also some good public schools. Elementary it is easier to find a decent public school. The best public middle schools and highschools are magnets, which means they are difficult to get into but not impossible. Depending on the part of the city and the lifestyle you would like to lead also will determine the answer to your question. Diverse does not begin to describe the Jacksonville sprawl. Different areas of town will feel different.
If you want to live at the beach, Jacksonville Beach elementary is a good option for public schools, not really sure about the middle school or highschool (most of my friends go to private school at that point if they are not in a magnate program). Bolles Beaches would be the top private school there.
If you want to live "over the ditch" (intracoastal waterway) Then Chet's Creek Elementary or J. Allen Axson Montessori are good public options (Axson is a magnate off Hodges Blvd). Private school would be Providence School which is PreK -12 and where a lot of students go after they are finished at Chet's if they do not get into the public magnate program.
Southside I am not sure of so I will not direct you there. I am sure there are good ones but most people I know in that area drive their kids to one of the private schools.
San Marco (close to downtown) has a good public elementary school. Hendricks Elementary (Do not confuse it with the private school called Hendricks Day School). Private schools in that area would be Riverside Elementary or Bolles K-12 for lower school and Bolles K-12, Episcopal or Bishop Kenny for High school. There are a lot of other private schools in the area but those are arguably considered the top.
Northside (Alta drive and Oceanway areas) New Berlin and Louis Sheffield are good elementary schools.
Do not know anything about westside except that I have heard St. Marks is a good private school over there. Hope someone else can lend their expertise to that area.
Also Orange Park...don't know anything about that area. Hopefully someone chimes in there.