'ci' translates roughly to "us" in this context.. vediamo is understood that you (the person and I am going to see) and I are the "we" who will see eachother tomorrow. Although I guess it would work for a group of people as well but "we see you tomorrow" is a pretty awkward way of saying that. Maybe a native Italian speaker will come along and tell us if 'ci vediamo' is used for groups as well as for two individuals. For instance, how would you close a meeting that will meet again tomorrow..

Shall implies futurity (determination, promise, or prophecy); will implies volition. There are subtle differences between when you use 'will' and 'shall,' and it's by no means accurate to say that 'will' is not first-person singular and plural in English - especially since every single native-speaker of English uses 'will' thousands of times a day to indicate a desire or plan to do a thing. We use 'shall' when we wish to say a bit more forcefully that something absolutely is going to happen. We use 'will' when we want to say a bit more forcefully that it's my desire and intention to do a thing.

Child: I will NOT go to bed! Parent: You SHALL go to bed - and right now!

It's a 'rule' almost not followed at all nowadays, certainly in ordinary speech, but here's Strunk and White (notice they are talking about formal writing, not spoken English - despite having as an example an apparently drowning person using formal written speech to invoke assistance): "Shall, Will. In formal writing, the future tense requires shall for the first person, will for the second and third. The formula to express the speaker's belief regarding a future action or state is I shall; I will expresses determination or consent. A swimmer in distress cries, "I shall drown; no one will save me!" A suicide puts it the other way: "I will drown; no one shall save me!" In relaxed speech, however, the words shall and will are seldom used precisely; our ear guides us or fails to guide us, as the case may be, and we are quite likely to drown when we want to survive and survive when we want to drown."