1. buy good quality strings, it's worth the money.
2. Go real slow when you tighten them. Give them even a minute here and there to accommodate to the climate, then go real slow again. it should work.
I've only broken a new cello string once in my life, a G-string I'd bought from a folk-music store. I go very slowly tuning up a new string, tightening it slightly higher in pitch over a period of several days or longer. This string was extremely difficult to tighten every day of the regimen. The folk-music store sold it to me without a bag or a label so I couldn't check this (they take all of their strings out of the bags when they arrive from the whole-sale stores then throw out the bags, wrappers, and labels), but I'm pretty convinced the string they'd sold me was not for a full-size cello. I can't think of any other explanation. Even extremely old cello strings don't break very easily--although some do break from time to time. My advice is to check the label, and don't buy cello-strings from any store that doesn't specialize in classical music.