A courteous approach
If they accept, perhaps you'll become friends, comrades, or rivals. If they refuse... don't take it personally. Perhaps their social circle is already too large for them; perhaps they wish to reserve some time for their Protégés.
Becoming Acquainted with someone increases Influence by 1; when your Influence passes 90, you can no longer add Acquaintances until your social circle has been reduced in size.
In Messages Tab
You will receive:
You have sent a Calling Card to [friend's name].
Your friend will receive:
[Your name] has invited you to accept their Calling Card: a polite overture to other social actions! Accept, and they'll become an Acquaintance.