In my experience, students in sophomore or junior level math courses usually have very little trouble picking up LaTeX on their own. They typically require the following assistance:
Some guidance in downloading and installing it, e.g. links to user-friendly distributions for both the Mac and the PC. I post links to my course webpage.
A sample LaTeX source file, 1-2 pages in length, using simple commands, and similar in format to a homework assignment for the course. Again, I just make this downloadable from my course webpage.
A relatively short first homework assignment. Most of the first "homework" is actually just getting LaTeX working!
Occasional help with LaTeX during office hours.
I have required LaTeX for all written homework when teaching introductory proofs, introductory abstract algebra, introductory point-set topology, and introductory real analysis, and in each case there was essentially no problem. Usually one or two students per class had trouble getting it up and running before the first weekly homework assignment was due, but they would all have it working by the end of the second week.
I really think that LaTeX helps students to write better proofs. It makes them spend more time on the writing, and it also gives them the ability to their edit proofs a bit after they write a first draft. Students often show me a LaTeX version of a proof during office my hours, and I suggest ways in which the writing could be improved.
Another huge advantage of requiring LaTeX is that students can simply e-mail you their homework assignments. I have a tablet and a stylus that I use for grading, so I just write my comments directly on the PDF and then e-mail it back to them. In addition to being faster, this has the nice advantage that you retain copies of the students' assignments, which can be helpful in a variety of situations. Without a tablet, it would work just fine to simply print out the students' assignments and hand them back as paper.
Of course, requiring LaTeX is somewhat more work for the students. Fortunately, most of the extra work for the students happens during the first few weeks of class, which means that it isn't much of a burden for them. I've never had students complain about the extra work.
One more note: I don't think it would work very well to require LaTeX in a course such as graph theory where students commonly have to draw pictures on their homework assignments. Even in topology, I had to drop the LaTeX requirement near the end of the course when we started talking about geometric topology, because the students' homeworks were full of pictures!