I can obvious just say the internet is fast/slow, but I want some different expression when it is/becomes super-fast/slow . (something like the internet flies? Is there a proper way of saying that?)
I don't believe there's a standard phrase to refer to fluctuations in internet speed (aside from the technical terminology used by internet service providers, but that doesn't seem to be what you're asking for).
As such, any word which conveys speed (or a lack thereof) could be used. I'm more familiar with such a word being an adjective though (rather than a verb, as you've used). Also, I usually hear references to one's own internet connection described as 'my internet' rather than 'the internet'.
My internet's crawling tonight.
My internet's flying!
However, people will often just use
My internet's being really slow/fast.
as you mentioned.