Currently, I encountered some weird sentences. They are: "She went off on some errand."(from Collins) "We watched him plough on, northwards at incredible speed, until he vanished from sight."(from Good English 10-Frankenstein) "Obtain considerable proficiency."(from my teacher) I am really confused with this and wondering how do you native speakers think about that.
"A[n]" is used as a determiner. In "some errand", the word "some" is also a determiner. The first example sentence "She went off on some errand." could have been expressed "She went off on an errand.", but the writer chose "some" instead, to intentionally express that the exact nature of the errand was of no interest.
Your next example, "sight", might have been expressed as "our sight", but "a sight" wouldn't make any sense. "Sight", in that use, is non-countable.
Your last example, "Obtain considerable proficiency.", is not an idiomatic sentence as you have quoted it. If the sentence were "You must continue to study until you have gained [a] considerable proficiency.", then "a" would be optional.