In a course about Turing machines, how many questions like "Construct a Turing Machine that does X" should I include?
I integrate many, many simple TMs into both my lecture and as homework.  I also force them to use descriptors such as $\vdash$ and $\vdash^*$ on their machines.  I've found that it takes a long time for students to get used to basic operations.  

They have no problem converting a given TM to its tuple members ($\delta$, $\Gamma$, $\Sigma$, etc.), but have surprising trouble seeing the performance implications of transitions on machines that I provide to them.  With practice, they get much better at this, but it takes (in my experience) a surprising amount of practice, both in encoding and decoding simple Turing Machines.

Some particular traps for my students have been: 


navigating halting states (since they cannot transition outwards)
identifying ending conditions of loops
making comparisons between two strings stored on different parts of the tape
learning when it is appropriate to overwrite your data.


All of these take practice.  I do this in-class so that I can assist and guide, but I also have many hours of facetime with my students.  In a more constrained timeline, I would give out homework assignments instead.