When mikveh night falls out following Rosh HaShanah, there are a few options for
preparation: (1) You could leave preparations for after Rosh HaShanah. (2) You could bathe and perform full preparations erev Rosh HaShanah, and try to keep the preparations as intact as possible. After Rosh HaShanah, you would shower, comb your hair, clean your teeth, remove any makeup, dirt, or other
chatzitzot, and inspect yourself thoroughly. (3) You could perform a few, more time consuming preparations before Rosh HaShanah (e.g., preparing your nails), but leaving the bulk of preparation for afterwards.
Given that there is often a bit of a rush at the mikveh after a holiday and that women are often fatigued at that point, we tend to favor the second option.
If you have a thorough bath before Yom Tov with mikveh preparation in mind, you need not take another bath afterwards, though in such a case you should still shower before immersion. (In general, a bath is not absolutely obligatory as part of mikveh preparation so long as a woman cleans herself very thoroughly.) You should
not bathe or perform any preparations on Yom Tov itself.
Please write back with any further questions.
Ketivah vachatimah tovah!