London Gatwick Airport: Take a direct Southern or Thameslink train to Brighton. It is the clear choice: roughly 25–35 minutes to Brighton station, usually about £10–20 one way depending on ticket and time. Do not pay for a private transfer; Gatwick is unusually well connected to Brighton by rail.
London Heathrow Airport: Use the direct National Express coach to Brighton Pool Valley/Coach Station when the timetable suits: roughly 2¼–3 hours, commonly £10–30 if booked ahead. Rail via central London is usually faster only if connections line up, but means a cross-London transfer with luggage.
Brighton railway station: This is the arrival station you want, not London Road (Brighton). It is a 10–15 minute downhill walk to North Laine, the Lanes, the Pavilion and the seafront; take a Brighton & Hove bus or a short taxi only for Hove, Kemp Town, the Marina, or luggage-heavy arrivals. Direct trains from London Victoria or London Bridge take about an hour; Thameslink also links St Pancras, Blackfriars, Luton and Cambridge.