replacing shifter cable housing?
Its likely that your bad shifting is due to messed up cables or a misadjusted derailleur. 

You can either cut the cable crimp off at the end of the cable with a pair of pliers, or pull it off with a pair of pliers. 

As for replacing the cable housing, you can either get your bike shop to cut a piece of housing of the right length by taking your old housing to them, and cleaning up the cut with a file + awl so its flat and the hole is completely open, or you can buy housing kits from companies like Jagwire and cut it yourself (but you will need a bicycle specific cable+housing cutter like Park Tool CN-10 or a dremel to cut the housing properly). There is a nice tutorial on how to replace your housing here. Another good page on prepping housing is here. 

As for replacing just the housing vs replacing the housing + the cable, its generally a good idea to just replace the housing + the cable, just because its not all that much more expensive and if your cable has some rust on it or has been damaged, it may lead to poor shifting even with new housing. I always replace the cable and housing together, just to avoid problems. 

EDIT: I zoomed in on the picture. The cable is toast. Get a new cable.