Birmingham Airport: Broadway is a car destination, so hire a car at BHX if you can: roughly 50–65 minutes via the M5/A46/A44, depending on traffic. By public transport, take the train into Birmingham New Street, then a Cotswold Line train toward Evesham and finish by pre-booked taxi; allow about 2½–3 hours and roughly £35–£60 before the taxi. Do not expect a seamless airport-to-village bus.
Evesham railway station: The most practical rail arrival. Evesham is on the Cotswold Line, with direct trains from London Paddington, Oxford and Worcester. Pre-book a taxi for the final 25–30 minutes to Broadway; expect roughly £30–£40. There are local buses, but they are useful only if their sparse timetable happens to suit you—not after a train with luggage.
Honeybourne railway station: Closer than Evesham and on the same Cotswold Line, but it is a minor unstaffed stop with no dependable taxi rank. Arrange a taxi in advance for the 15–20 minute ride to Broadway, roughly £20–£30. It is a good choice for cyclists and hikers, not for improvising an arrival.
Moreton-in-Marsh railway station: Useful only if you are collecting a car or have a taxi arranged: Broadway is across the steep Cotswold escarpment and there is no direct bus. A taxi takes about 20–25 minutes and costs roughly £30–£40. Do not choose it assuming the village is an easy walk or casual bike ride with bags.
Cheltenham Spa railway station / Cheltenham Racecourse: For the Gloucestershire Warwickshire Steam Railway experience, take Stagecoach Park & Ride D between Cheltenham Spa and Cheltenham Racecourse, then the heritage railway to Broadway when it is operating. It is scenic leisure transport, not an efficient airport-style transfer: the bus stops are at opposite ends of the large racecourse, with about a 20-minute walk between them, and the steam railway does not connect to the mainline.