I think you mean the Hack Green Nuclear Bunker. It's now converted to a museum. It's at Whitchurch which is about an hour and a half from Kidderminster, about an hour from Bridgnorth.
If you like a pool with slides, waves, sprays etc then Crystal Leisure Centre at Stourbridge isn't too far away.
Ironbridge is good for a day out - avaoid a Sunday afternoon as it is completely full with tourists at this time of year. As someone mentioned, there is a pass to get in all of the museums that lets you go in each one and lasts a year. Otherwise, Blists Hill in Telford is lovely, full of character. Love the old sweet shop and the old fairground rides. It's set up like a working town, with a bank, and shops etc.
Black Country Museum is in Dudley. Similar, and I love it there too. Same thing - little shops, good fish & chip shop etc. Your tickets allows you a return visit too.
Personally I wouldn't do Dudley Zoo, it's hilly and over rated. If you want to see animals, do the West Midland Safari Park in Kidderminster. You need the best part of a day to do the safari and all the rides.
There are lots of little historic places in Shropshire to see - I know most of them because of school trips in the past. Just need an hour of so at each of them.
Wroxeter Roman City is nice as you can visit the reconstructed house and the remains of the bath house. A bit like an English Pompeii.
Of towns to visit, if I only did a couple I would do Ironbridge (standing by that historic bridge is great) and Shrewsbury - it has lots of listed buildings and churches (The Abbey is where the Brother Carfael series was set and filmed)
Much Wenlock is a cute little town for a look around as well.