The "six months" refers to how long before your passport > expires < and not how long you have had it. Your passport is valid for travel the day you receive it.
Citizens of about 38 countries can travel to the USA for up to 90 days at a time as a tourist or for some business purposes by registering for ESTA > before < their trip begins. The cost is US$14, which is similar to the fee for the Australian ETA program (though a bit more than the Canadian eTA program).
Registering is accomplished online, you don't need the details of your trip, and the response is usually immediate. The confirmation is valid for multiple trips to the USA over two (2) years so don't wait until the last minute. VERIFY the information you enter is correct. For example, human-error clicking Yes to the question about ever-before overstaying or otherwise violating the terms of a USA visa will cause your application to be denied. Correcting the error will be difficult and time consuming. You may even be forced to apply for a regular visa, which is expensive in addition to being time consuming and difficult.
The online application has a question about arrests and convictions. It ONLY applies to major crimes and drug arrests/convictions as an adult (age 18 or older). Answer NO to the question unless your personal history includes an event which counts as a major crime committed as an adult or drug offense committed as an adult. UK Police Cautions are not issued for disqualifying crimes, so answer No if you received one. Exception: Get legal advice if you received a UK Police Caution for illegal drugs as an adult.
Print the confirmation - or at least record the number - and keep the information with your passport. Having a return or onward ticket leaving North America is a requirement, so have a copy of your airline itinerary at passport control.
Check with your health insurance to find out if it covers you in the USA and pays for medical evacuation to home (should that be necessary). If yes then take proof of coverage with you. If no or not sure then get trip medical insurance, which is cheap and sold by airlines & travel agents. The USA has wonderful medical care but it isn't free or even cheap. Please do not skimp on this as an otherwise-silly accident could turn into a crisis if you don't have medical insurance. This is a link from the US government, but the info is good for anyone traveling outside their home country:
If you are planning a long visit then DON'T schedule your departure day for the 90th day (any part of a day in North America counts as a whole day, so if you arrive at 23:59h on the 1st and leave at 00:01h on the 10th it is 10 days in the USA). Schedule your departure not later than the 88th day as there are no (zero) provisions in the Visa Waiver Program for someone to voluntarily extend their stay. If you miss your flight due to something Immigration might consider avoidable (such as a flat tire on the way to the airport) they may ban you from using the VWP in the future.
If you are a minor (age 17 or under) traveling unaccompanied to visit friends or family in the USA then you need a notarized letter of permission from your parents which gives the name(s), address, and phone number of the full adult(s) who will be responsible for you in the USA. The letter should also authorize the adult(s) in the USA to act on your parents’ behalf in case of medical emergency.