Heathrow to Brighton- need to book ahead?

No need to buy ahead, London - Brighton is commuter distance, just turn up, frequent trains per hour.

Don’t take Heathrow Express, take the Piccadilly underground line to Barons Court station, make an easy few feet platform change to the District line take this to Victoria station, follow signs to the larger National Rail station and buy your Brighton ticket there if you couldn’t at LHR.

(Not certain if you can buy the whole journey to Brighton ticket at LHR)

See London train and underground map below., follow the route I’ve mentioned

https://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf

Or, there is a National Express bus that goes to Brighton from Heathrow without any changes:

https://book.nationalexpress.com/coach/#/choose-journey

It bypasses London.

Thanks so much!