Jewellery Quarter is you haven't been - just a fascinating place to see. I think there is a little museum there. Lots of fashionable places to eat
Digbeth's Custard Factory - quirky / arty / eateries
Wander around the City Centre - Victoria Square is where the City's 'Council House' is, and Town Hall, and there are quite often things going on in the square there. Also the Art Gallery (which has been closed for refurbishment but I think will be open by then) which is free and worth a look round even if you don't know much about art. Also has nice Edwardian Tea Rooms in it. Then walk through to Centenary Square which is nicely done, and it is worth going in to what we all call 'the New Library' which I think is lovely, but as with all new buildings, not everyone agrees. Go up to the Shakespeare room. Go out on to the viewing gallery.
Then walk round to Brindley Place, walk along the canal for a bit, or take a trip on one of the canal boats. Lots of eateries there, not too pricey.
We have St Philips Cathedral (Anglican), St Chad's (Catholic) and St Paul's Square is a nice place with lots of eateries - not far from Jewellery Quarter if you don't eat there. St Martin's in the Bull Ring is the oldest Church though.
We have a Chinese Quarter if that is your thing.
Theatres - there is The Hippodrome, The Alex and The Rep which might be places to look if you want to see a show.
@SleepyHedgehog 's ideas are good too, and there is also a station in the heart of Bournville if you don't want to walk so far. Cadbury World is there, which locals think is a bit expensive for what it is, but there are usually offers with train tickets or with clubcard points and I think worth a visit if you haven't been before. If you are in Bournville, the carillon is often played on a Saturday afternoon if you can make that work. Have a look on-line for the history of Bournville - built as a model village by George Cadbury.
Or, if you are in to sport, you'd have to look up fixtures but we have cricket at Edgbaston (bit early in April), Football at Aston Villa, Birmingham City, West Brom, Walsall just a short tram or tain ride away. Often athletics meets at Perry Barr and various sports happen at the University.
I don't think you can beat a visit to the Nature Centre at Cannon Hill Park (simple bus ride out of town in Edgbaston) - now known as Birmingham Wildlife Conservation Park. I go there without having any dc to take, every now and then, as do many people I know. It really is lovely and excellent vfm.