Birmingham isn't a massive tourist destination, but there are a few things in the city and nearby that may be of interest.
I spent 3 years in the city at university (Aston Uni, in the city centre). Aston Uni main building is apparently the world's largest building built with small red bricks. It has a couple of "Skylifts" on the outside - basically lifts going to the top floors with plastic windows.
There is of course shopping in the city centre, along with the city art gallery.
There are apparently more canals than Venice, and whilst most are filthy and contain shopping trolleys, some are quite pleasant to walk along, especially around the Brindley Place district, full of waterside cafes, bars and restaurants, and the National Sea Life Centre aquarium. A walk from here down towards University can be pleasant, especially for a transport buff as the canal follows a railway line.
In Bournville is Cadbury World, a large chocolate factory and museum, with tours and big shop - take a local train to Bournville.
The Jewellery Quarter is apparently famous for jewellery shops, but I wasn't impressed.
Or take a train to Stratford, home of Shakespeare, about 40 mins from the city on the end of a suburban line. Another line goes to Warwick, home of the impressive castle.
If you like steam trains, take a local train to Kidderminster, where the terminus of the Severn Valley Steam Railway is.
A little further afield is Ironbridge - take a train to Telford, about an hour away, then a bus onwards to the site.
The Bass brewery at Burton, another destination 40 mins or so away, can also be visited and toured.
Hope this helps!
Flights booked: NWI-AMS-JNB-DUR, JNB-AMS-NWI