The easiest way by public transport is to get the train from Birmingham to Manchester Piccadilly and then get the Metro link which runs from inside the station to Old Trafford. The Metro drops you half a mile from the ground. There is a rail station at the ground but it is only open on match days.
If you book the train tickets now online you should be able to get cheap advanced price tickets.
This FA site tells you the bus numbers from Piccadily to Old Trafford. You could get a National Express bus from Birmingham which would take you to Chorlton Street Bus Station in Manchester. This is close to Piccadilly for the Old Trafford buses.
http://www.thefa.com/England/SeniorTeam/NewsAn dFeatures/Postings/2004/07/PoliceAdvice_Travel lingFans.htm