You do all your preparations on Erev Rosh HaShanah. Pay special attention to cutting your nails and combing your hair, as those are things you cannot repeat on Yom Tov. Keep your hair tied back over Yom Tov to prevent as many tangles as possible, and try to avoid sticky substances (don’t worry, honey washes off easily!) and situations that could result in chatzitzot. You may also wish to avoid heavy or hard-to-remove makeup. Over the course of the next day, try to stay away from potential chatzitzot that will be difficult to wash off with just liquid soap and water. You can eat normally on the holiday, including meat.
You begin your final preparations on the night of your immersion after tzet hakochavim (when the stars come out, the first day of Yom Tov is over, and you can start preparing for the second day). Since it is Yom Tov and not Shabbat, you can wash one part of your body at a time with warm water, liquid soap, and no wash cloth. You should also clean your teeth as permitted on Yom Tov. There are several opinions about how to do this. We follow the position that you use liquid toothpaste (not gel) and pre-cut dental floss. If flossing would definitely cause your gums to bleed, then use toothpicks or a rubber tip instead of floss.
The time of immersion will be after dark on the second night; check the exact time with your local mikveh. At the mikveh, run your fingers gently through your hair to remove tangles, and check yourself to assure that there are no chatzitzot. You wet yourself immediately prior to immersion with one extra dunk in the mikveh. You may want to read the article Immersion on Shabbat and Yom Tov on this site for additional information.