Good, and consistently so. Nine in ten graded settings in the council area are rated good or better. Two are outstanding. None is failing. Schools aren't what stops a family moving to Hungerford. The asking price on the house near them is.
Ofsted has graded 23 settings across West Berkshire, and 21 of them are good or better. That's nearly all of them. Two are outstanding, 19 good and 2 require improvement, with none inadequate. These counts cover the whole local authority, not only Hungerford and its villages, so they describe the system your children would enter rather than grading the school at the end of your road. Another 42 settings hold no published grade yet. They haven't been judged either way, and nothing about them should be read into that silence.
The names in this answer tell you what exists nearby, not where your child will get a place. Admission runs on the council's rules, and those rules don't care which school you drove past on the viewing. Check the admissions criteria for the specific school before you exchange, not after. Families who skip that step learn about the rules from the offer letter.
In town, Hungerford Primary School carries the place's name. At secondary level, John O'Gaunt School takes its name from the fourteenth-century lord of the manor who gave the town its fishing rights on the Kennet. It's one of the few schools in the country named after a landlord people still thank. Out in the villages, Kintbury St Mary's Church of England Primary School and Inkpen Primary School serve their own communities, and there are small primaries at Lambourn, Chilton Foliat and Shefford too. Oaklands School completes the local list. For the most part this is village-school country: small primaries, one market-town secondary, and a school run that often involves a car.
The schools are one of Hungerford's real strengths. Nearly everything inspected is good or better, so the harder part of the decision is the house.