how would I know which C to play, and if the notation showed a 'C' to a 'A' note, how would I know which A to play? QUOTE]
Your confusion is understandable because the bass has multiple positions and strings at which one can produce the same pitch. If you are in standard tuning, when reading a transposed bass part you see "Middle C". Well, you can play that these ways:
5th Fret=G String
10th Fret=D string
15th Fret=A string
20th Fret=E string
(*notice a very important pattern here? ...down (or up, depending on how you look at it) one string, over 5 frets*) This is how our fretboard repeats...
Average bass players in general have a comfort zone on the neck and they like to stick around that area. Most players would just play it on the 5th Fret G string, since this is the home base, meat and potatos area of the neck. But, often in styles like funk, jazz, and latin there are octave leaps that would require a huge shift if you played it down there, so it would be best to play it on a different string higher up the neck. If you observe funk and latin bassists you will see that they often occupy the middle of the neck (ie 5th thru 11th Fret more or less)more often than their rock counterparts, because those grooves use root to octave leaps all the time.
Bass players with real reading chops read ahead and scan the piece to see where the highest and lowest notes are. This allows them to best choose which position to start from so that they don't get caught off guard and lose the groove because a large, unnecessary shift trips them up.
Also, there is another lesser consideration to think about when choosing where to play a certain pitch: TONE. The E string is a much larger, more massive piece of metal than the G string, so a "Middle C" on the E string has different tonal properties than a "Middle C" on the G string. This is something you should explore for yourself.
To truly be comfortable, competent, and confident reading for the bass you gotta get out of your comfort zone and really learn to read in those areas of the neck that trip you up. Good luck and welcome to the best clef.