The Lickey/Waesley Hill round is my from the door ride. Surprisingly nice and green considering you are only 11ish miles out of town. As others said Cannock/Wyre are not too far away.
Selected parts of Rubery ( +1 for the canal into town, under 1 hour easy, 10-15 mins on a few minor road). Alvechurch? Pricey, but nice.
Mosley and Harbourne if you want a midweek drink. A cheaper option a few years back now could be Kings Heath, you can ride across a few parks to get to the Rea/Canal route into town/uni.
Alternatively, you could rent overlooking Edgebaston Res, thinking you are in Edgebaston… and actually be in Ladywood.
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