Glad you want to visit Edinburgh! It is a beautiful city. You could get the train and the journey lasts about 5-6 hours with probably 1 change. You can find times on http://www.thetrainline.com or http://www.virgintrains.co.uk.

You could also get the bus which takes longer, 8-9 hours at http://www.nationalexpress.co.uk Most of the airlines fly between Edinburgh & Manchester and you could also try http://www.jet2.com I don't know anything about them except that they are cheaper!

Enjoy Edinburgh! Jan Scotland (near Edinburgh!)