With the first, I'm fairly sure I'll have a good relationship with you all. With the 2nd, you're slightly more sure that you'll have a good relationship with them. So, first sentence, I *think* it'll be a good relationship, the 2nd sentence is a little more certain that it will be. Keep in mind though, that stress, intonation can change the meaning completely. If I said, "I EXPECT (stressing this word)..." you are almost demanding that this relationship will be good. ie. Telling whoever you are speaking to that they better make the relationship work !

It's difficult to say without context. The use of 'anticipate' might be friendlier than 'expect' but it would depend on context, who the speaker was and their personality as well as who they were talking to and the relationship.

I anticipate having an enjoyable working relationship with you all. I expect (to have) an enjoyable working relationship with you all. I look forward to having an enjoyable working relationship with you all. I reality most people would use these interchangeably, but 'anticipate' is more formal, and the meaning is slightly different. It implies more that you expect it, or regard it as a forgone conclusion. Look forward to is more natural to the native ear.