Correct, each block is one stitch and you do them according to color. If you change colors you'll need to make note or redo the chart to reflect those new colors.
You start a new color by just knitting with it. There will be a gap there, but you can weave the end in after a few rows to close it. After that you loosely carry the colors up the side. I twist them once to keep them tighter, but you may or may not need to do that. It depends on how far you're carrying them and since it's a tube it won't show.
Oh and circular knitting is a spiral so you won't need to do anything to start the next row. You might want to put a marker though so you know where it begins and ends. I put the marker between the second to last stitch and the last one. That way you've always knit past it at the end of a round and it won't fall off.