Are you familiar enough with the general workings of InDesign?
I cannot download the Avery template so this is pure conjecture. However, I think your problem may be that the template contains indicators (placeholders) as non-printing items or on non-printing layer (or even both), and you are not supposed to change these. If you do, these items will show up on your pages!
Open the original template again and carefully inspect the Layers palette. Is there more than one layer? If Avery thought this through, they saved it with at least an additional "normal" layer for you to work on.
If there is only one, is it marked "non-printing"? If so, add a new layer (and make sure it's not set to "non-printing", obviously) and draw your text on that.
Also, check if you are not accidentally working on a master page. Stuff that you put on a master page will not appear in the 'regular' document, unless you apply that master page to a 'regular' page. (And it's not the recommended workflow either.) All your own text should appear on your own pages.
Furthermore, no need to send it to your printer to see what happens! Export to PDF and look what appears on your screen! There is a checkbox in the Export dialog to automatically open the PDF when it's finished exporting. Don't forget to close the PDF, though, else you cannot overwrite it for a new attempt.
Do not be tempted to check the box to force printing of "Non-printing objects"! This will bite you back at some point; either you get everything you asked for, plus all guides and whatnot which you did not ask for, or you decide to switch it off and discover that you placed some of the stuff you want to be printed on non-printing layers.
Finally: use the various preview screen modes in InDesign to your advantage. Do your design with all guides and marges etc. visible, toggle to Preview mode to see how it looks without them. InDesign is smart enough to automatically hide non-printing object when in preview mode, and it should look just like when finally printing.