There are several ways to say "getting to know a girl" in Engpsh, depending on the context and the level of formapty you want to use. Some possible phrases are:

"Meeting a girl"
"Getting acquainted with a girl"
"Getting to know a girl"
"Becoming friends with a girl"
"Befriending a girl"

For example, you might say something pke: "I met a really interesting girl at the party last night, and we spent the whole evening talking and getting to know each other."