"I am going to the library."
Translation:Idę do biblioteki.
"Jadę" means that you are taking some vehicle. Which is a perfectly fine interpretation of "I am going", so added.
"Idę" does mean that you are going by foot, although frankly it can be used also if you are planning to use a vehicle. I don't know where to set the border though. "Idę do sklepu" if I am taking a car is okay to me. "Idę do Ani" if I am taking a bus as well. So is "Idę do biblioteki" although I'm using a bicycle.
But well, you wouldn't say "Idę do Krakowa" unless you literally go there by foot. I also wouldn't say "Idę do centrum" unless I lived close enough to literally walk there. So maybe it's safer to just use "idę" and "jadę" literally...
"na" + Accusative also indicates movement towards, but it's only correct with some places, just like "do" + Genitive is only correct with some places.
Rule of thumb: "na" is used with open spaces, "do" with closed ones. There will be numerous exceptions though, I'm afraid.
Anyway, firstly library should take "do", and even if it took "na", then it would be "na bibliotekę".