Birmingham is smaller, a lot less affluent and has much less public investment than London - much like everywhere else in England!
As a result, it has a lot more empty shopfronts and lot fewer independents than London. Not that it's not worth visiting but if you want something that isn’t just the same chain stores and coffee shops as everywhere else, you do need to go looking.
The back to backs on Hurst Street are a national trust property, showing how most people lived in Birmingham before the centre was bombed during WWII. Well worth a visit but needs prebooking- or at least the tours do, which are well worth it.
The Coffin works on Fleet Street is interestind as an example of the kind of small metal processing businesses that underpinned the economy of the city for centuries. The factory was operational until the 1990s and is interesting also because of the story of the last owner, a woman who started as secretary and worked her way up, facing considerable sexism. The surrounding area of the jewellery Quarter is also interesting.
If you want independent, arty Birmingham, then the Custard factory down in Digbeth has bars and independent stores. Last time I was down there they were still building the tramway, and the pavement was all dug up, but apparently that's done now although the tram won't be running for a couple of years due to delays to HS2.
Grand Central is just another name for New Street station. It's got some shops, but not many more than you'd find at somewhere like King's Cross. It's not really a shopping destination.
Bullring is the big shopping centre by New Street station. You can access by the NSS exit close to Leon. It's fine. It's a shopping centre full of chain stores and restaurants. But I don't think you'd find anything in there that you wouldn't find in most districts of central London.
The canal area around the library is nice. The ikon gallery, across the canal, is the main modern art gallery in the centre and worth a visit if you're in the area.