What is the Catholic Church's teaching on magicians?  It is permissible to watch a magic show, or to even become a magician?

Now, what I say "magician" I do not mean actual magic, such as witches, warlocks, or anything paranormal.  I mean illusionists, who do things such as card tricks or other things that appear supernatural, but which everyone understands is not.

Magic shows in general would not pose any moral problems.  But some of the magicians today clearly incorporate the occult into their shows.  These are the ones that would be morally objectionable.

If you like magic, here is a pretty cool act (with no references to the occult):

peter15and1 - I think that denzinger 1653 and 1654 answers your magicians question.

The Misuse of Magnetism*

[From the Encyclical of the Holy Office, Aug. 4, 1856]

1653  . . Already some responses on this subject have been given by the Holy See to particular cases, in which those experiments are condemned as illicit which are arranged for a purpose not natural, not honest, and not attained by proper means; therefore, in similar cases it was decreed on Wednesday, April 21, 1841: "The use of magnetism, as it is explained, is not permitted." Similarly, the Sacred Congregation decreed that certain books stubbornly disseminating errors of this kind should be condemned. But because, aside from particular cases, the use of magnetism in general had to be considered, by way of a rule therefore it was so stated on Wednesday, July 28, 1847: "When all error, soothsaying, explicit or implicit invocation of the demon is removed, the use of magnetism, i.e., the mere act of employing physical media otherwise licit, is not morally forbidden, provided it does not tend to an illicit end or to one that is in any manner evil. However, the application of principles and purely physical means to things and effects truly supernatural, in order to explain them physically, is nothing but deception altogether illicit and heretical."

1654 Although by this general decree the lawfulness and unlawfulness in the use or misuse of magnetism were satisfactorily explained, nevertheless the wickedness of men grew to such an extent that neglecting the legitimate study of the science, pursuing rather the curious, with great loss to souls and detriment to civil society itself, they boast that they have discovered the principle of foretelling and divining. Thus, girls with the tricks of sleepwalking and of clear-gazing, as they call it, carried away by delusions and gestures not always modest, proclaim that they see the invisible, and they pretend with rash boldness to hold talks even about religion, to evoke the souls of the dead, to receive answers, to reveal the unknown and the distant, and to practice other superstitious things of that sort, intending to acquire great gain for themselves and for their masters through their divining. Therefore, in all these, whatever art or illusion they employ, since physical media are used for unnatural effects, there is deception altogether illicit and heretical, and a scandal against honesty of morals.*

Delores - The OP can likely deduce from it.  The sentence I see of importance is this -  

"When all error, soothsaying, explicit or implicit invocation of the demon is removed, the use of magnetism, i.e., the mere act of employing physical media otherwise licit, is not morally forbidden, provided it does not tend to an illicit end or to one that is in any manner evil".  

The "illusionist art" is specifically mentioned at the end, and it speaks of the unnatural(magic included) throughout.  What the OP needs to discern is whether any of it is "illicit" or not.  

In my opinion, if the man advertises himself as a magician, or one performing a magic act, then I would consider it illicit regardless of the details(major or minor).  But, if the guy advertises himself as an illusionist strictly adhering to acts simply of illusion(physical media), and avoids all that has to do with magic/magicians/the unnatural in both word and deed, then it would seem to be not illicit according to this docuмent.  

But, how likely is that?

I say, "suffer the children to be fed".  Labor for the peace and building up of the church.  Avoid near occasions.