Hi nikki,
I'm not sure what sort of cycling you're after but,
As Veloman says, Sutton Park is a good fun option. Take the bike on the Cross City Line to errm, Sutton.
However, if you do this at the w/e, you'd be well advised to travel early in the day as the trains do get busy later in the day.
You could ride to the city centre using the Rea Valley Route, or even better, use the Rea Valley or canal route to either Bournville or Selly Oak and take the train from there. This has the advantage of you already smugly occupying the cycle space on the train as it arrives at New Street where all the other bikes and hundreds of baby buggies pile in.
Alternatively the are some lovely lanes to the south of Brum. To get out of Brum you will have to climb. This is unavoidable, but once you are up there are some lovely routes.
A suggestion for escaping the city could be Rea Valley/Canal to Primrose Hill. (As you huff and puff up Primrose Hill just keep thinking how your climbing is improving and how much fun you're going to have when you come back down.