I started using Mathematica at age 60. I have to say that becoming fluent in Mathematica (then not yet called the Wolfram Language) was one of the hardest, and most fun, technical challenges I've ever undertaken. Learning to express my not-so-great existing math understanding in this rather mysterious Lisp-like language was a huge amount of work.
Today, with the maturity that comes with turning 64 (the old-age equivalent of becoming a teenager) I can say that the payoff has been worth three or four times the effort. As a result of looking things up in Wikipedia and trying them in Mathematica, I've become familiar with ideas I would have found unapproachable on my own.
As an example, consider the concept of Gaussian curvature. This interesting formula for measuring the stretched-ness of a surface at a given point illustrates why one can bend a sheet of paper in one direction with perfect ease, but cannot do so in two directions at once without stretching the sheet -- which is pretty much impossible with paper. Now, Saran Wrap -- that is stretchable. After reading the article in Wikipedia, I easily coded my own gaussianCurvature function, and put it to good use in pursuit of gluing together two or more pieces of paper along lines that curve in the (bent!) planes of the sheets.
As if that were not interesting enough, it turns out that the very same concept is used in General Relativity to check whether spacetime is curved in a way that creates internal stresses in our 3-dimensional materials, known as tidal forces. Of course it helped a little (!) to watch a few hours of Leonard Susskind's superb introduction to General Relativity from Stanford University on youtube.
And here I must disagree with some other answers and comments about not using Mathematica "too much." Granted, I may not be made of the same strong stuff from which real mathematicians are built. But for me, the more I use Mathematica, the more math I learn.
Edit: I notice that I have not answered your question, but... yes, I think you can make great use of Mathematica as long as you don't get overwhelmed by the challenges of really learning it, deeply. That should not happen, but if it ever does, please feel free to email me, firstname.lastname@example.org, and I will be happy to offer any help I might be able to give.