Kids with you, or kids at heart? Bourneville has the Cadbury chocolate factory next to the station, with Cadbury's Chocolate World tour, just over 10 minutes from New Street.
Stratford-upon-Avon, just about an hour from Snow Hill, say no more. But did you know it also has a butterfly farm with loads of huge butterflies flying about the exhibit and landing on or near you? Or a restaurant on a narrow boat right in the centre?
Kidderminster, 40 minutes from Snow Hill, has the Severn Valley Railway which runs frequent preserved steam trains on 16 miles of scenic route through rural Worcestershire and Shropshire countryside to Bridgnorth. Side bonus is Captain Cod's over the road from the SVR station in Kidderminster, where you can have very good fish and chips at a budget price.
Worcester, a little beyond Kidderminster, with its world famous cathedral and bells (make sure you get off at Worcester Foregate Street, not Shrub Hill)
At an hour and 10 minutes from New Street is Oxford with its dreaming spires and colleges, a wonderful botanic garden, and everything you've ever heard about it. Bonus, San Francisco style burritos.
Lots of canals in and around Birmingham and all the towns and cities mentioned above. Have a ride on a narrowboat in Gas Street Basin in Birmingham.
And who said Birmingham is boring?
enjoy your journey