Is it right, why 'with'?
I suggest you join with me.
At least in contemporary English, the phrase join with is used to mean to attach. This sentence probably does not mean "I suggest you attach yourself to me," so either the sentence is very old or they meant to say one of these two:
• I suggest you join in with me.
• I suggest you join me.
The sentence in the Question is an incomplete idea. It lacks context. It is impossible to be sure what the writer intended.
Nevertheless, "Join with me" and "join me" do not necessarily mean the same thing. "Join me while I drink this pint of beer" is an invitation to watch me drink it. 'Join with me in a drink of beer' implies that we shall all be drinking beer.
"Join with me" implies a request to take part in some collective activity. "Join me" on its own might mean that but it need not.