Hi Ceri
You need to do a spatial join,joining up your trails to your fishnet layer. Right-click on fishnet layer, Joins and Relates > Joins ... Ensure you select the "Join data from another layer based on spatial location" option, and "Each polygon will be given the numeric attributes of the lines ..." Ensure Sum is checked on.
If you are in a geodatabase, then just use the Shape_Length field. Otherwise you may need to first calculate feature length of the shapefile segments following your intersect.