The screen you are initially confronted with is a mostly blank page with a series of tilde characters down the left side of the window and a filename at the bottom left corner. The tilde characters represent the end of your text buffer; you will see the column of tilde characters get pushed farther down the screen as you begin to fill the buffer.
The initial mode of vi is command mode. To begin editing an existing document you can stay in command mode, but if you wish to start a new document you must switch to text-entry mode. To do this, press either 'i','a','o', or 'O'. The character 'i' selects insert mode, which causes characters to be inserted at the current cursor position. 'a' selects append mode, which moves the cursor one character to the right before inserting text. 'o' and 'O' stand for open mode; selecting the lower/uppercase letter will ''open'' an empty line after/before the current line and place the text cursor at the beginning of this line. The important thing to remember is that all of these modes are equivalent after the initial selection of the mode. All modes act as insertion mode after they perform their initially different behaviors.
After selecting a text-insertion mode, you can begin typing your document. It is important to realize that the document that you are editing exists only in memory as a text-buffer, it won't be placed on disk unless you explicitly put it there. This feature allows you to completely screw up a document, then back out of the problem by exiting without saving the document. If you have made editing changes that you don't wish to save, then you can type :q! to throw the current buffer away, leaving the document on disk untouched. If for some reason the system crashes while you are in vi, you will receive mail from the system explaining that if you issue vi -r <filename> you can recover your edit session, with generally no loss at all.