Question: My understanding is that Masters and PhD students often need to TA -- even if they're already research assistants to a professor. 

I'm a first year undergraduate who took courses heavily TA-reliant last semester. My impression is that my TAs (oh they were helpful people) don't really get much out of TAing. The job just covers some of their costs and their tuition.

Sure, you may learn a bit about teaching others, but I doubt that you may learn much after three weeks of doing so. Many of the users on this site were once or are TAs. Do you agree with my view?

If TAing is really a necessary evil, are there better ways to fund your education?

Answer: 
It may be necessary; some universities require students to TA even if they have other sources of funding. It need not be evil. It depends on your attitude (do you want to help students learn?) and which faculty you are working with (Do they care if you help students? Do they care if you learn something yourself?). Unfortunately you usually do not get to chose who you work with. TAs (and also faculty) should receive training in how to teach, but may not.

There are many other ways to fund education, some of which pay better than being a TA, and some of which are more prestigious. They all have limited supply.
I understand that it might often be thought that one learns little more after TA'ing for a few weeks, or a year, or a few years.

However, I strongly disagree. I have seen no cases where anyone came to a profound understanding of the psychology of 18/20-year-olds even after years of dealing with them. Nor understanding the complicated, self-contradictory goals of lower-division mathematics. Nor... Many research-oriented people in math never do quite catch on, although the norm is an uneasy truce with the seemingly-ineffable realities.

I truly think that until one can nearly-effortlessly do a bit (of course, hours within some limits...) of TA'ing, one has not understood the situation of academic mathematicians, and does not understand the job itself. (The "research" aspect has its own invidious pitfalls, but/and these are fairly different, perhaps opposite, from those of TA'ing.)