British Nationals can stay in the Bahamas for up to 21 days, however this can be extended up to a maximum of 8 months by applying to the Department of Immigration in Nassau. If you are travelling to the Bahamas via the USA, you will be required to apply for a USA ESTA, and any time spent in the Bahamas will count towards the 90-day maximum permitted stay in the USA.
Your passport must be valid for 6 months from the date of departure from the Bahamas. All visitors leaving the Bahamas are subject to a departure tax, which may or may not be included in the price of your ticket. You should check with your airline to confirm this.