The art of the Cheese and Pickle sandwich is a subtle one, but easily learned.
First: The bread. Do not use supermarket-bought pre-sliced bread, it is flimsy and has abolutely no flavour. The bread needs to have some bite to it, some resistance to the teeth. My personal preferance is for a crusty granary roll, but this is very much a personal matter.
Use unsalted butter, which has been out of the fridge for at least half an hour. This is abolutely vital, as it will not spread otherwise. Use just enough to seal the surfaces of the bread, it is very easy to overdo.
Cheese: this should be a mature farmhouse Cheddar, double Gloucester or Wensleydale. Cheap, generic "mild cheese" will not do: a bit of acidity is necessary to achieve the correct flavour. It also cannot be said enough: American cheddar is not Cheddar. Contrary to what mrichich says above, it should be sliced about 3mm thick, not shredded.
As far as pickle is concerned, I have only one thing to say: Branston. Accept no substitutes. Only the original Branston pickle has the right balance of sugar and acid, the right crunchiness. That having been said, don't overdo it: the cheese should definitely be the major component, and the pickle is there to complement it.
Avoid supermarket own-brand "sandwich" pickles, they just don't cut the mustard.