Meet vs. Meet with vs. Meet up

Meet
Meet and Meet with mean the same thing.They can be interchanged in some sentences, but there are some differences in usage and meaning between them :
1-Meet is much more common than the other two.
It means that you encounter a person by the chance:

- He's the kindest person I've ever met.
- I was walking down the street which I met john.

2- Meet is used when you are introduced to a person:

- Yesterday, I met her father in the cinema.
- I 'm sure you wish to meet the great writer.

3- It could simply mean that's where we are going to see each other and we're going to do some work together:

- They sometimes meet each other for dinner and chat.
- We could meet for a drink after walking.
- Do you want to go see the new movie at the theater?
Yes! I'll meet you there in ten minutes.

Have a meeting with
It means come together to discuss. This could be a substitution for meet and meet with.Have a meeting with is usually used in formal writing.

Meet with
Meet with means " having a meeting or appointment with someone or coming to discuss " It is an American expression which is now used in many other English speaking nations.It is more formal than meet .So it is used in business,negotiations, .... it implies that it has very professional reasons .

- You will meet with the managing director tomorrow .
- The president is going to meet with the Chancellor at nine o'clock.

When we encounter a person for the first them, we meet them.

- Have you met my new friend?
- I wish to meet the great singer someday.

Meet up with
Meet up is an informal and even near slang expression which means "to meet someone for a specific purpose."

- The two of us will go shopping tonight,and we will meet with you later.
- Perhaps we four could meet up for dinner?
- Twice a week I meet up with a group of other young mothers for a baby afternoon and --.
- Subject: Re: Meet up at Spurs game.
- We 're going to meet up for a cup of coffee.

Also, NTC's dictionary of phrasal verbs and other idiomatic verbal phrases‎
by Richard A. Spears - Language Arts & Disciplines - 1993 - 873 pages
Page 470 :
meet up with someone or something means to meet someone or something, usually by accident. (Colloquial.)

Notice the comparison :
- Meet with me for lunch tomorrow.
- Meet up with me for lunch tomorrow.(common)
- Meet me for lunch tomorrow.(more common)

Here is a comment from The American Heritage Dictionary :

Meet with is a Standard idiom (I hope to meet with them next week), meaning “to hold a meeting, an encounter,” as distinguished from I hope to meet them, which could also mean “to be introduced to them.” Meet up (I hope you and I meet up again some day) is Casual at best,possibly dialectal. Standard users would use only meet at higher levels. Meet up with, however, is at least Conversational, and at least one dictionary considers it Standard; it simply means “encounter,” as in I hope to meet up with her again some day.

References

BBC learning English.com
Wordrefference.com