Just got a 3/4 cello. How delicate are cello strings compared to guitar strings? I heard the A string breaks easily.

I have had my cello since April, and I have had my G string break once. That's the only one so far. I rent my cello, and I don't think they put new strings on when they gave it to me, so they seem to be holding up fairly well so far.

If you don't abuse the string then it is not likely to break. At least not for a couple of years. I would advise you to make sure your face is not in the firing line, because if they do break due to old age then they tend to break at the tail piece end.

The strings should last you at least a year + half the size
A = 1
D = 1 1/2
G = 2
C = 2 1/2 -> 3

If you don't go overboard with the tuning pegs you should be fine XD