http://tickets.eastcoast.co.uk/eight/
31 Jan.
Edinburgh - York depart 0800 price £8.00 single
From York, you have various options, such as TransPennine Express to Warrington Central, (walk through Warrington, you could grab a bite to eat from a shop on your way), Warrington Bank Quay to Birmingham on Virgin.
You could change at Manchester for a Northern train to Crewe, then Crewe to Birmingham on London Midland.
You dould take East Coast to Peterborough, then XC 170 from there to Birmingham.
But what I'd recommend, is taking an East Coast service to Doncaster, a TPE 185 to Sheffield, then a HST to Derby, and a XC 170 from there to Birmingham. If you want walk-on tickets, get a ticket to Sheffield, then a Wayfarer (York station are happy to sell this), then a Burton upon Trent to Birmingham. If you can book in advance, you can get a XC Single from York to Sheffield for £7 and there is always good availability on that route.
If you want to take Northern from York to Sheffield, take the 11:01 via Pontefract Baghill. (York - Moorthorpe £9.60 + Moorthorpe - Sheffield £4.00 is the cheapest split for walk-on single tickets from York to Sheffield on trains that call at Moorthorpe).