When used with a noun, the adjective “first” is almost always preceded by “the” or a possessive determiner, such as “my”, “your”, “his”, etc. For example:
This is my first wife.
Neil Armstrong was the first person on the Moon.
When used in its literal sense of “the first time of seeing something”, “at the first sight” is not an exception. You can say, for example:
However, “at first sight” is much more commonly used in the idiom “love at first sight”, which means “an instantaneous/immediate attraction”. It is used without an article in this idiomatic meaning:
“Love at first sight” is the most common way to employ this phrase, but “at first sight” can also be used in the sense of “based on the first impression”:
Again, there is no article when the phrase is used in this sense. Another variant of this idiom is “at first glance”, which follows the same rule.