The schools are a large part of why Chandler's Ford exists in the shape it does. This is a family suburb, and family suburbs are built on catchment. There are infant and junior schools scattered across the estates, two big secondaries doing the heavy lifting, and colleges within reach for the sixth-form years. What this answer does not have in front of it is a stack of official inspection grades to rank them by. So take it as a map of what is here, not a league table of which one wins.
At the younger end, the town runs on the old split model of separate infant and junior schools as much as all-through primaries. Fryern Infant School and Fryern Junior School sit as a pair, as do Hiltingbury Infant School and Hiltingbury Junior School, so a child can change buildings without changing neighbourhood. Chandler's Ford Infant School, Merdon Junior School, Scantabout Primary School and Knightwood Primary School fill in the rest of the estates, with St Swithun Wells Catholic Primary School covering the faith end of the choice. It is a lot of provision for one suburb, which is exactly what you would expect somewhere this heavily weighted toward houses and families rather than flats and renters.
At secondary, the two names that come up on every school run are Thornden School and The Toynbee School, and they are the reason a good number of families buy here in the first place and then refuse to move for two decades. Beyond sixteen, nearby colleges handle the sixth-form and further-education years without anyone needing to send a teenager far.
Here is the part the postcode will not tell you honestly: living near a school is not the same as getting into it. Places are allocated on admissions criteria and distance, not on which estate you happen to like the look of, and the catchment lines move. Buying a house because a particular secondary is up the road is a plan, not a guarantee, and the families who treat it as a guarantee are the ones opening a thin envelope in the spring and wondering what went wrong. If a specific school is the whole reason you are moving, confirm the current admissions rules for that school before you confirm anything with an estate agent.
The blunt verdict is that Chandler's Ford gives you plenty of schools and plenty of choice, and that is genuinely worth something in a place this size. What it does not give you, in this answer, is a ranking, because the inspection grades are not here to hand you one. Read the latest report for whichever school matters to you and judge it yourself. On the numbers available, the schooling here is a solid, unremarkable middle: a real reason to consider the town, not a reason to skip the homework.