Winterbourne gardens are in Edgbaston and worth a visit. The Botanical gardens are also round the corner. I prefer Winterbourne. You could combine with a stroll through the university campus (lovely actually) and there is the geological museum on campus (Lapworth). The museum is free. Winterbourne isnt.
The canals are a big part of Birmingham heritage. You could walk the canal from the Vale on campus. You could take a canal boat tour or the water taxi (cheaper). The museum and art gallery are free in the city centre. There's a nice edwardian tea room in there for a cuppa. Check the opening hours tho.
Further out the Lickey hills for a stomping country park style walk. Car park charges.
You could drive to Bournville for a mooch about. The green is really pretty. You can go into the Cadbury world shop without paying admission. Not lots to do tho really.
The university train station will get you into Central brum in a couple of stops.
The jewellery quarter draws a lot of visitors and there are various museums, pubs and cafes and all of the jewellery/gold shops. Bit dull if youre not shopping imo.
Finally cannon Hill park and the MAC (Midlands Art Centre) is nice. Car park charges apply. There's gallery spaces, makers markets, an independent cinema and theatre spaces. There's a cafe. Check what's on. This is a really really short drive from Edgbaston.
City centre wise There's the cathedral, the library, the museum and art gallery (already mentioned), the bull ring and all the usual city centre stuff.
Is this enough to go on?