I am looking at a piece of music and there is a C# in the key signature. On the cello there is a C note on the G string and on the A string. I know that in a C# key signature the C on the A string is a C# but I want to know if the C on the G string is a C# as well.
Yes, if the key signature included C#, it affects ALL C notes. No need to look for esoteric cases where a composer may have invented his own private system where something that LOOKS like a key signature has a non-standard meaning, though I'm sure some could be found. A key signature affects all octaves.
I guess there's an F# in the key signature too?
The position of the sharps and flats on the key signatures is traditional and, generally, seems to have been "designed" to fit within the stave. If C# is indicated in the key signature then all C notes are sharp unless indicated otherwise during the piece.
On the cello you can play a C# on one or more strings - depending on the pitch. Which one you use will depend on the left-hand position and the preceding and following notes. There will be a difference in tone though and this will be more and more detectable with experience.
There must be an F# at the beginning followed by C# so we're on the key D. Whenever you see a C note, it means to play it in sharp unless you see a natural sign. And when you see such, it will only be effective for 1 bar.