Well there are two ways over the Rea Brook from the housing estate you live in - one path follows the old Bridgnorth railway line and goes from Sutton Road itself, towards the Darwin public house. The other leads from Laundry Lane (I think this is the one you mean). Both end up at the Sutton Lane/Park housing estate. The shorter route is the one that leads from Laundry Lane and I believe it goes under Pritchard Way too, saving crossing that road.
You can cross over the railway line (at Sutton Bridge Junction) between Sutton Lane/Scott Street and Betton Street. It was once a road, but has been pedestrianised. You'll see what I mean when you're there.
At the bottom of Betton Street you'll be in Coleham. From here you can cut through Egland Lane (a shut*) and find yourself on Longden Coleham. From here you can cross the river at Greyfriars' Bridge (when exiting Egland Lane, turn left), which takes you over to the town centre at St Julian's Friars.
Easy! :D
*shuts are passages in Shrewsbury which cut from one street to another. Egland Lane has a pub on either end (one on Belle Vue Road and another on Longden Coleham). More info about this shut can be found here -
http://shrewsburyforum.co.uk/index.php?showtopic=440(scroll about a third down the page for Egland Lane) :)