Cannon Hill Park is about 1 mile from the centre, follow signs for the MAC or Edgebaston Cricket Ground, it’s opposite the latter.
Or, take your bike and go and ride some urban stuff, there are lots of steps and drops to play on, just follow your nose and you’ll find something. The canal is a good place to start as is the The Mailbox.
If you decide a pub may be your best bet a few to try are:
The Post Office Vault – 350 beers to try in a great little dirty bar.
MPW @ Mailbox – A bit pretentious but has great views over the city.
The Actress and Bishop – Over towards the Jewelry Quarter, a bit of a dive but good music and a great atmosphere
The Old Contemptible’s – A Birmingham institution, good beer, my Dad’s favourite pub!
There are loads more, if you want some more ideas post back!
And if you’re not drinking just take a wander around, it’s really quite a nice city. Head to the Mailbox for a coffee maybe, walk around the canal’s towards Brindley Place, the area around Snowhill Station and towards St Paul’s sqaure is nice and the lounge at Hotel du Vin is not a bad place if you’re no drinking.