As an educator in engineering myself. I have to say...I'm a bit offended at the question really. But not by you, to make things clear.
I'm not really sure where it's coming from...
To answer your broad question,
You should learn what you want to learn...It was the entire point of your degree...you studied engineering, because it's what you wanted to do for yourself, because it's what interested you, because machine design, and the practical use of mathematics and sciences are fun and interesting and challenging all at once.
Well, I guess that's how it is supposed to be in my idealised world. Of course, a lot of people are just at university because it's the thing to do after high school and engineering has prestige and a high wage.
To answer your question specifically, yes.
There are no pure mechanical machines being used in industry anymore (well I'm sure there are some exceptions) and it's rare that one can or would bother to calculate anything by hand anymore for anything serious. Programming, mathematics, CAD and other computer related engineering software in general are all modern requirements today.
You should start spending a couple hours learning some basic things that are programming related, design related, or well, related to your job if you don't know anything that they expect of you...Assuming you want to do well in your position.
Simply waiting for them to tell you what to do, sets yourself up as a not particularly interesting employee. A manager will be far more impressed in someone who strives for personal (and indirectly company) improvement, than someone who didn't bother to do a little of their homework before coming in on their first day.
As a suggestion, I would recommend learning your personal favourite math software (matlab, mathematica, maple as examples), python, java or c++, and some form of CAD software (inventor, solidworks).
I (and any good manager as far as I'm concerned) would take an average worker who is personally invested than a lazy genius who can't be bothered.