Colour changes are normally worked on Right Side rows - this makes the colour change look clean and neat.
When its time to change colours, drop the first colour and, leaving a long tail, knit a stitch with the new colour.
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The first stitch will be very loose. That is OK, after you work a few stitches in the new colour, you can tighten the stitch by gently pulling on the tail.
Continue working in your new colour, leaving your old colour where you dropped it.
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If you are working stripes in only a few colours and with a stripes only a few inches wide, you can carry your colours up the side of your work. What this means is you do not have to break the yarn each time, just drop the yarn you have been working with, bring the ball end of the new yarn around under the old yarn to wrap it in place and start knitting. This means a lot less ends to weave in.
When it comes time to weave in the ends, use a tapestry needle and work the loose ends through a number of the purl bumps on the back of the work. Don't work in a straight line, rather work in a number of different directions to better secure the yarn.