^ Wow, you must have a killer GPA if they offered you to enter the PhD program right away! Congrats!!
Judging by your posts, you'll have the best time of your life in grad school. You definitely sound like a dedicated, uber-motivated math genius
To answer your question, I did teach for 2 years at the college/undergrad level (History & Latin) while finishing grad school. It was a rewarding experience that made me realise that teaching is not for me. Even though I
love preparing lectures, choosing class materials, improving my knowledge of the subject I'm teaching, seeing my students get hooked on the subject I'm discussing, etc., I found that I simply can't handle the pressure and the stress that come with teaching. Preparation, reading, grading, answering students' emails, all that requires a HUGE time investment, even more than writing your thesis. And grading can easily become overwhelming if you can't hire TAs because of budget restrictions.
Playing the «authoritative figure» role on a daily basis was the hardest part for me. I tend to sympathize with students, and they appreciate it, but sympathy can complicate a lot of things... For an example, some students should have failed my classes, but I couldn't bring myself to grade them honestly.
I will always remember what happened after I gave a D- to a few students during my first semester of teaching. One of them made an appointment with me to discuss his grade, and started crying loudly (!) as soon as he walked in my office. I was mortified, I didn't know how to handle it, what the hell do you say to an adult man who is crying because of a failed exam?
He was the first one to react that way, but he was certainly not the last - I saw students walking out of other teachers' offices, crying, after every single exam that semester. It was a very uncomfortable situation. It forces you to play the «psychologist» role, and I suck at that. So that's why I probably won't teach again.
But I'm addicted to research! That's what I've always wanted to do, as far as I remember, as a kid I used to take notes on every single thing I read, even comic books, haha... i had all these little notebooks filled with misc info and I added something every day... Right now I have a part-time librarian job that I like very much, and I use my spare time to get involved in research projects in my field. Most of the time I get paid for it, but when I don't, I still try to get something out of it by submitting articles to journals or sending conference proposals around. So far it has worked pretty well, I feel so lucky to have that oppportunity.
Sorry for the rambling btw, I guess I'm in a nostalgic mood today! I can already say that grad school was the best time of my life. It made me a better person. It definitely changed my personality in a good way.