Cut the cabbage in half and peel off a few of the thin outer layers. Cut each half into quarters, then cut out the tough core in the middle. Slice each quarter crosswise into thin shreds, or run the quarters through a food processor with a shredding blade. Pour the shredded cabbage into a large bowl.
Peel the carrots, then either cut them into very small matchsticks, or shred them using a food processor with a shredding blade or on a box grater. Transfer the shredded carrots to the mixing bowl with the cabbage.
In a smaller bowl, make dressing by combining mayonnaise, vinegar, celery seed, sugar, salt and pepper (if desired). Toss dressing into cabbage mixture and let chill.
If you need to make the coleslaw a day or two ahead, I'd shred the cabbage and carrots, and then store them undressed in a container in the fridge until you need them. The dressing can be made in a mason jar or other small container and when you're ready to assemble the salad, just shake up the jar and pour it over the veggies. The longer you store the finished coleslaw the soggier it will get.
