Do I need to let red wine breathe each time I uncork the same bottle?
Some wines improve with aeration.  Others, not so much. Mostly reds, but deciding which is which is a matter of experience and some information.  Many younger red wines will benefit from, at least, swirling in the glass...but could also improve in a decanter left open for 30 minutes to an hour.  I would guess that most people are drinking wines that are less than 5 years old.  Many of these wines improve with some air.  Older wines is where experience and knowledge come into play, as some need air and others will fade with time exposed.  

Saving is another story.  Red wine should be vacuum sealed and refrigerated, as air will continue to work on the product.  ...over time, this is not necessarily a good thing.