I've had a teacher flatly tell me that とっても is incorrect, but I do see it written here and there and I'm pretty sure I hear it as well. Is it just so informal relative to とても that I should never use it in a classroom or on schoolwork?
とっても is a spoken variant of とても, just like すんごい is a spoken variant of すごい and あんまり is a spoken variant of あまり. If you're writing a paper or speaking in a formal setting, it's better to use とても.
とっても is just a strong form of とても according to 大辞泉, so both are correct.
It has similar pronunciation (may be a bit different intonation) with 取っても, so may be it could confuse some.
But usage of とっても can be found since 昭和３０年（1955) (at least) from this song called
So, I believe you can use it most of the time but if the place you are going to use demand too formal, you may need to use non-strong one とても. But I doubt most people would not care or even notice it unless they are tracking you like in your case.
It's like the difference between really and reeeeally. とっても is not incorrect, but it might be a little colloquial to use in a classroom, depending on the teacher. It's even in my dictionary with a couple of examples. In the dictionary it says 「とても」を強めていう語, it makes とても stronger.
とってもおいしい - reeeallly tasty!
だめだなんて僕にはとっても言えない - There's no way I could say that!
Generally, とっても gives a stronger emphasis to what you're describing, but it can be "too strong" in certain situations, such as formal and semi-formal situations(even with new friends) and real writing.
On the other hand, it's generally A-OK to use とても in any situation, though for more formal situations, you might want to substitute the word out with 大変（たいへん）.