While dunking once in the mikveh is halachically sufficient, there are various explanations for the practice to immerse three times. Some explain that, on a technical level, multiple immersions ensure that at least one is valid. Other sources cite the fact that the root "tahor" is found three times in a verse in the Torah (Leviticus 15:13) that relates to the mitzvah of immersion. Dunking three times is one of a number of different customs.
Showering at home after bathing is sufficient, as we see from immersion on a Friday night when it is generally prohibited to shower again at the mikveh. However, it is customary to shower again at the mikveh, as well as recombing hair and visually checking the body, as there is a risk that a chatzitzah (barrier) could adhere to the body or hair while one is dressing and traveling to the mikveh.