Most likely you will shampoo your hair anyway on erev Yom Tov. But with such a long gap between erev Rosh Hashanah (Wednesday) and motzei Shabbat, you should shampoo again after Shabbat. You will certainly need to comb through your hair again. (On an ordinary motzei Shabbat, if the hair did not get dirty, it would be sufficient to shampoo your hair on Friday and just wet it with warm water and comb it before immersion.)

You should bathe in warm water after Shabbat (paying special attention to crevices and folds of the body), even if you bathed before Yom Tov. You will need to brush and floss your teeth after Shabbat. To save time at the mikveh, certain more durable preparations, such as cutting nails, can be done before the holiday.

As with every tevilah, you must examine yourself carefully for chatzitzot just before you immerse.