At BHX, stay airside—limited amenities but efficient flow favors staying put.
Take the Air-Rail Link monorail to Birmingham International, then the train to New Street (10 minutes total from monorail, GBP 3.60). Walk out of New Street into the city center. The Bullring and its bronze bull statue are immediately south. Walk north 10 minutes to Victoria Square and Chamberlain Square for the town hall and civic buildings. Head back at the 90-minute mark.
Follow the New Street route to Chamberlain Square, then visit the Museum and Art Gallery (free, 1 hour). It holds one of the best Pre-Raphaelite painting collections in the world. Walk 15 minutes northwest to the Jewellery Quarter: a still-operating district of workshops and indie food spots. Lunch at any of the cafes on Vyse Street. Return via New Street.
With 10 hours, take the train one stop further to Cadbury World in Bournville (GBP 23, book in advance). The train from New Street takes 15 minutes to Bournville station. The tour covers the history of Cadbury and includes generous chocolate samples. Alternatively, the Thinktank science museum near Digbeth (GBP 17, 1.5 hours) works well.