This looks like a fairly simple pattern to me. Heck, Berroco even calls it an "easy pattern." So that's good!!
You use the two different size needles to knit different sections. In this case (and for many garments with ribbing), you do the ribbing in the smaller size needles. Then when you're ready to move to the next stitch pattern (probably stockinette in this pattern), you switch to your larger size needles. Often it's faster (and perfectly okay) to knit the first row of stitches off
the smaller size needle using the larger needle (so the last row of ribbing would be in your left hand on the smaller needles; then you would take one of the larger needles in your right hand and knit knit knit away until all stitches are knit onto the larger needle. Of course, you would need to use the other large needle to complete the next and subsequent rows).
Here's my suggestions: Do a gauge swatch (or two or three) to see what size needles you should use to get gauge with your preferred yarn. Wash and dry the swatch per the recommended directions on the ball band and make sure the yarn you've chosen isn't going to misbehave itself (shrink or grow, for instance). Remeasure for your gauge (to make sure you are still getting the right size). If your gauge is off, do another swatch and wash/dry it, adjusting accordingly based on the results of your first wash (I'm assuming you WILL want to wash the vest).
When your gauge swatch is spot-on, print this pattern out, mark or highlight all the numbers for the size you want to make. And then begin! If you take this one step at a time, you can do it.
Plus, if you get stuck, you can always come here and ask folks for help. They always have a solution (usually two or three).