3.5 to 4 ounce skein should work for a scarf; you need some yarn for fringe, too. Could get 2 skeins so may have extra if you need it, depending on how long you go, length of fringe, etc.. It is best to get your extra up front so you know the yarn color matches (colors sometimes vary from different dye batches). Get 4 ply & check for skein stating washable. When find that area of yarn & types, feel the softness of each for your choice. Avoid extra 'fuzzy' types of yarn: the extra fuzzy types don't look nice after washing & difficult to crochet for a beginner.

Small 'holes' give more warmth so a single crochet stitch would be tighter. I prefer a double crochet stitch for air circulation & if wanted it thicker, I would make it twice as wide (then have option of folding it when putting on or creating a seam so it has 'two sides').

Get a set with various hooks. Small hooks makes small stitches and larger hooks makes large/looser stitches.

F, G, H, I, J, K works with 4 ply. Find which hook feels comfortable for you. I use the 'H' or 'I' a lot for scarf, afghan, stocking cap.

Regarding width of scarf, do a practice with 11 slip knot then one row of crocheting. The # you start with will depend on hook size & type of stitch used. If it is too short, pull it apart & start over. The scarf will look better after you had a couple of practice rows first and the hook with yarn more relaxed in your hand - - inside of 'freeze'/ tight in start & loose/relaxed on the rest.

The length will be determined as you move along. You WILL know where you wish to stop.

Fringe. Length in inches you want X 2 + 1 & cut. Example:
3" fringe: 3 X 2 is 6. plus 1 equals 7. Cut 7" yarn strips. Take 2 & poke middle into an end stitch, the loop that comes thru is where you put the threads into. Pull tight. You now have 4 fringes in one end stitch. Skip one or two stitches & repeat. When done, trim unmatching fringes.

Have extra yarn, make a matching stocking cap. Length from top of head to forehead is your 'width' & continue as if making scarf. When long enough to comfortably reach around your head, do a seam. Now it looks like a cylinder. Weave a yarn thread in and out on one of the cylinder edges and pull tight = that is top of head. Pom Pom = wrap yarn about 75 times around object size of cigarette pack. Take a thread about 8" long & tie around chunk of those wrapped threads, tie 2-3 knots. Slip yarn off the object you wrapped around & holding the tied part. The loops hanging down, cut through so they are 'fringe'. Take your knotted thread (8" one) & pull each of the 2 threads into top of cap (each of the 2 threads in separate cap holes) & tie underside of cap.
Trim pom pom threads so they are even.