cat piped to grep can check to see if a line of text contains a string. the sed command is more suited for editing files. Sed is a good utility to modify text. Sed is good at deleting strings and also substituing one string for an other.
Since the forward slash has special meaning, you need to escape it. By placing a backslash in front of the forward slash, all of the special meaning is removed(escaped). for example my home directory is /home/shane. I will do an example that works
shane@mainbox ~ $ pwd | sed "s/\/home\/shane/\/new\/directory/"