初めてお目にかかります。

English Translation

I am glad to make your acquaintance.

Does this sentence literally mean, “It’s the first time I have set my eyes upon you”? It sounds very formal.

@ericaw
初めてお目にかかります is a business formal version of 初めまして (= nice to meet you) when you make an acquaintance of someone.
If you prefer a word-for-word translation, it means “It’s the first time I, as a grubby person, am within your range of sight.” 目にかかる literally means “X hangs on Y’s eyes”.
Your proposed translation:
“It’s the first time I have set my eyes upon you”
is not correct because you swapped the subject and object, and thereby it sounds arrogant in Japanese. The speaker is humbly saying this phrase by using a passive form.