What "soft-skills" does one need to succeed as a mathematician? My question is about "soft-skills", i.e. knowing how to handle Latex, giving a speech or giving a lecture, or other knowledge that could be useful for a career in mathematics.

Skills that (all) mathematicians need, but which are not necessarily something that one learns studying the mathematics of text books or lectures.

This knowledge should therefore benefit mathematicians of varying disciplines and not only the algebraic topologists or the number theorists.
A few that come to mind, and are probably valid in other fields also -- 


Learn how to give good talks.
Learn how to describe your research informally. Say you meet someone at a conference, and they ask you "What do you work on?" Can you convince them that what you work on is interesting?
Learn how to come up with interesting questions that you don't know how to answer.
Learn how to read a paper. Can you figure out "the big idea" without getting bogged down in the technicalities?
Learn how to meaningfully participate in a math discussion when you're less than 100% sure what's going on.
Develop a gut feeling for when a subject will come in useful. For example, "This lemma feels like I could use algebraic geometry to prove it", without knowing initially what that form would take.
Get a sense for what other researchers in your field consider interesting. What is worth writing a paper about, and what is worth giving a talk about?


There are many, many more -- and none of the above are easy.