Nottaclue, I think what they are doing is asking to do just as you think. Finish the blue row. Let the blue yarn drop. Begin knitting with the white & finish that row. Cut the blue yarn, and leave at least a 6" tail. By not cutting the blue yarn until you have finished a row with the whie, I think maybe the first stitch with the white will be easier to do, instead of cutting the blue and having the last blue stitch loosen as you try to start the white row. This is the first I have heard a pattern stated this way, but thinking about it, that is my opinion.

Welcome back to the world of knitting!

Tami...who was just asked to teach a friend of DD's to knit!

Thanks for the advice! That is what I did, but I tied a knot, or sort of a half slip-knot, with the white around the blue, just to keep it from loosening as I started the white. I can easily untie it, if it looks too bad when I go to weave in the ends. But after all that, I saw a mistake I had made when I switched over to the white PLUS I, somehow wound up with 4 extra stitches, so now I have to rip it all out and try again. I think I got the extras because I wasn't as careful as I thought I was being when switching back and forth with K1 P1 (Ribbing stitch?). I am supposed to be bringing the yarn between the needles when switching, right? I know... silly newbie questions!!! Thanx again!!

nottaclue;

This is a blue and off-white striped scarf right?

To change the colours, just start knitting with the white yarn, INSTEAD of knitting with the blue yarn - the same as when you started your first row of knitting at the beginning. No "carrying" of the blue, or knitting with it. (Putting a slip knot is a good idea so the beginning stitch doesn't loosen up. At least until you get more experience with your tension.)
I think the reason the pattern says to cut the blue yarn is so you don't end up with a tangled mess when you start the next row of the new colour. So you can cut the blue at any time - and my preference would be to cut it as soon as you finish your last blue row.

Yes, K1 P1 is ribbing. And it is very easy to acquire extra stitches!
I was initially confused when I read "between the needles when switching" because I thought you meant switching the colours. But yes, bring the yarn between the needles - towards you - when switching from knit to purl. And between the needles - away from you - when changing from purl to knit.

Be sure to follow Tami's advice about leaving a 6" tail - of both colours. A "good" six inches.
That will give you enough yarn to tie a tight knot AND sew both ends in when you are finished.

Sorry for writing so much.
Hope this helps.