Are you importing the text, or typing it in?
Have tried both methods. First 3/4 of first page was previously imported via copy/paste. For the last 1/4 of the first page, there are two paras. When I pasted both, the second para disappeared until I deleted the first (which, in itself, exceeded the remaining space on the first page). When entering text manualhy, after page was filled, any further typing was not echoed to current or next page.
When typing or pasting, that's the way it works unless you have Smart Text Reflow enabled (I can't stand it so I leave it off). The text that does not fit is called "overset" and is inidacted by a red plus sign in the Out Port of the text frame. If you click the plus sign with the selection tool it will pick up the overset and you can place it into an new frame (which will be threaded to the original). Holding the shift key when placing the text will flow all that there is, making new frames and pages as required to accomodate it.
This is basic function of the program. You should spend some time reading the help files on working with text, or pick up a good beginners's book, like Sandy Cohen's Visual Quickstart Guide to InDesign.