Yes, I would plant your Avocado (Persea americana) in potting soil now. It is interesting to watch seeds and cuttings sprout roots in water, but planting in soil does save a step. Also, the skewer wounds on the seed could be a source of infection, but with any luck everything will turn out fine. If possible, grow it in a container over winter and then plant it in the ground in Spring.
An avocado will do best with full sun, regular water, and good drainage. Your hardiness zone of 9b-10a should be fine for most types of avocado. Do you know the name of the parent cultivar that the seed came from?
Down the road, when the tree is larger, you will want another variety for cross-pollination. Since this is a seedling it is unknown which varieties will be best for cross-pollination. Fruit quality is also unknown, but if it turns out to be very good, you might potentially have a new cultivar. I should add that self-pollination is possible, but not reliably so. You can try hand-pollinating flowers to give you greater yield.