Others have mentioned most of my favorite tools. One thing I'd like to add is SageMath. It's a mathematical software that's comparable/better than commercial offerings like Mathematica and MatLab. I've rarely seen anyone in the academia using anything else these days. If someone does use something else, it's just because they're more used to it. SageMath is by far the best tool for most things math.

Also, while typing about Sage, I was reminded of how great of a tool LaTeX is. If you want to write anything that'll be more than a single page, LaTeX is probably the best way to do it.