A word for first name and middle name/s, but not last name

given names

the name that is chosen for you at birth and is not your family name:

• Ex. Her family name is Smith and her given names are Mary Elizabeth.

Cambridge

Merriam Webster and Wikipedia tend to agree.

You might say forenames for first and middle names

forename - A person's first or ‘Christian’ name (OED)

In my lifetime, the two most common words used to describe names other than the family name are given names and Christian names. The latter has been steadily falling out of favor. Indeed, a google ngram view of the two shows exactly that phenomenon: n-gram of christian name, given name. Although, it should be noted, in common usage both labels are applied to mean the first name, or commonly used name (should it be other than the first) only. There is no single word, in English, that will automatically be understood by the recipient of a communication (verbal or written) to mean "all the names a person has not including their family name".

(Edit: for an interesting comparison, add forename to that n-gram view.)

Try personal names

: a name (as the praenomen or the forename) by which an individual is intimately known or designated and which may be displaced or supplemented by a surname, a cognomen, or a royal name

Merriam Webster