First, the easy part: "meet up with" is just an informal version of "meet with"; they mean the same thing. I would not use "meet up with" if I were talking in a formal situation, such as to my boss.
"Meet" is one of those words where the context really makes a difference. For example:
We will meet our son when he is born next month.
In this case, you haven't met your son yet, because he hasn't been born. So "meet" here means pretty clearly "meet for the first time".
I will meet my boyfriend's parents on Tuesday.
Here, the usual implication is that you don't already know your boyfriend's parents - that said, this is just an implication and may not actually be the case.
I will meet my friend at the airport on Tuesday.
In this case, I (presumably) have met my friend before - otherwise they wouldn't be my friend.
All of these uses of "meet" are valid. The context may make it clear whether you are meeting for the first time or not (or, the context may not make that clear).
"Meet with" has a slightly different implication. "Meet with" implies that there will be a "meeting" - in other words, you will be spending some time together discussing something.
If you are "meeting the President", that might only take 10 seconds. The President walks up to you, you introduce each other, and shake hands. There, you've met the President. But if you are "meeting with" the President, that implies that you and the President will be in a meeting together.