I'm trying to make a tagline for my website, can I say "download files at super fast speeds"? Is this grammatically correct?
Many speeds are recorded at (plus something)
"The Hong Kong Observatory put the storm's maximum ten-minute sustained winds at 260 km/h (160 mph) prior to landfall in the central ". (Wiki)
"She types at (a speed of) 56 words per minute."
A child's vocabulary grows at an average rate of (however many words per year depending on age.)
I won't go into 'at vs of', I will just say that for your purpose, your word choice is fine.
Yes, you can say it, and it is idiomatic (though I would write superfast as one word).
Another use of 'at speed' might be:
The crash happened at speed.
meaning the crash happened while (the car) was travelling at a significant speed. i.e. not while stationary or when maneuvering.