There are no rules about the width or length a winter scarf should be. Skinny knitted scarves have become quite fashionable in the last several years, and they are a fitting project for a new knitter. Using different patterns will cause the skinny scarf's appearance to change completely. Therefore, you can make many scarves using the same knitting needles, with many different looks.
Garter Stitch Pattern
The garter stitch is an appropriate first project for a beginner. After you've learned the knit stitch, there is nothing more to do than to knit every row. Still, you can get different looks with various thicknesses of yarn and different sizes of knitting needles.
Ribbing is generally seen at sweater collars, cuffs and bottoms. It's that stretchy portion of the sweater that hugs the body. It's also an interesting look for a skinny scarf. Rib patterns vary. Try several versions and see what appeals to you. Try the k (knit)1, p (purl) 1 pattern, the k2, p2 pattern or even the k4, p4 pattern.
The seed pattern, also called the moss stitch, will make small bumps in a skinny scarf pattern that resemble seeds. It's a simple pattern using two common knitting stitches, knit and purl. K1, p1 across the row. When you turn your work, use the same pattern but make sure you are using the knit stitch where there was a purl stitch in the row below. Make sure you are forming a purl stitch when there is a knit stitch in the row below.
Basket Weave Pattern
When casting on the stitches for a skinny scarf in the basket weave pattern, you use a multiple of eight plus five stitches. This will help you to follow the pattern without making errors later. This pattern is named basket weave because the end result looks like the weave in a basket. The pattern's first row is k5, p3, k5 across. The second row is p5, k3, p5. You repeat the pattern twice and then reverse the stitches. The second pattern is p5, k3, p5 on the first row across. The second row is k5, p3, k5 across, repeating the pattern twice. These eight rows make up the entire pattern. Repeat them until the skinny scarf is as long as you like.
Make any knit stitch pattern appear different by using more than one color. Make a pattern of two, three or five colors of yarn for a sharp contrast in your scarf. Use yarn left from other projects, changing yarn colors as you run out of each one. This will give varying widths to the stripes. Create a different kind of color pattern by using more than one strand of yarn at a time. Of course, you will need larger needles than you use for one strand. When this kind of coloring is used for making a scarf, it's best to use yarns with different textures. You will end up with a more interesting mix, but overall the scarf will hold the same color pattern, rather than decisive stripes.