Meet can mean two different but similar things. One is that you are introduced to someone for the first time. The second is simply that you went to a place at a time such that you and another person were there together when you had not previously been together. So you can meet someone new, then meet them the next day for lunch. However, if you "meet with" someone, you always mean the same place and time whether or not you previously had met them. An acquaintance is someone you know but not well enough that you are friends. So when you meet someone, you could say they are an acquaintance. You could just say you know him or her. If I had just met someone online, I would not say I made their "acquaintance". I would use "met" (as in met someone new). This is because I really don't know anything about that person, and I have not seen them in real life. Again, if you "meet with" someone online, you mean that you were at a specific time and place(website, instant messaging service, etc) together.

to make the acquaintance of = to come into social contact with

You don't really use acquaintance in many situations. Usually you say "pleased to meet you" when you meet them for the first time. After that it is just "good to see you". You do not need to say you are happy to meet someone.

get acquainted with That is the right expression.