@Aphorist - where are you based? In the UK it would probably very much be a side job, but I don't know what the situation would be in the USA.
As someone who works in post-16 education, I'd also like to say that the teacher probably did not have as easy a job as you think...
If you want to improve your public speaking skills, teaching will certainly help :-) I was quite shy when I started Lecturing, but after 10 years I am very comfortable with it and enjoy being the centre of attention.
What is your day job? What skillset do you have that would make you employable and enable you to pick up a couple of hours work teaching an evening class at a college? In the UK getting temp / part-time hours at a College is very much dependent on who you know: we don't tend to advertise these externally. So you need to do a bit of networking to get a class and see if you enjoy it.