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Kate's scarf looks like a basic garter stitch scarf, in which you'll knit every row until the length desired. Also looks like a bulky or heavy worsted weight wool yarn, probably knit on 13s or 15s. Here are a bunch of yarns you can choose from. I'm just guessing but I'd say cast on 15 stitches or so and knit, knit, knit. This would be a great beginner's project!
Once you pick the yarn you want to use, come back and I can tell you exactly what size needle and how many stitches to cast on.