Being in the centre of Liverpool, it's virtually impossible to avoid riding on main roads.
This is the route that I sometimes take towards Frodsham
http://www.gmap-pedometer.com/?r=3241326
Notice it avoids most main roads but you still have to ride on at least one main road.
Going northwards you can go through Bootle, Litherland and Maghull, that avoids a lot of main roads but you still have to ride on some.
Get an Ornance Survey map and have a look for possible routes.
But you just have to bite the bullet and ride on the main roads.