The Sefer Yesod Sion writes that scrupulously observing Shabbat, with all it detailed Halachot, has the power to atone for sins. As it says in the Gemara (Shabbat 118b) based on the Pasuk in Yeshaya, “Shomer Shabbat Me’halelo, V’Shomer Yado Me’Asot Kol Ra-One who keeps the Shabbat, guards himself from all iniquity,” meaning observing Shabbat will bring forgiveness even for idol worship like in the days of Enosh. Specifically, conducting oneself with holiness in the area of Shemirat HaBrit on Shabbat is a major Tikun.
He continues and writes that singing Zemirot at the Shabbat table with Kavana of (mentally) confessing one’s sins has the power to save one from the Din (judgement) of Gehenom. He then quotes from the Sefer Menahem Meshiv that this is alluded to in the Pasuk, “Mizmor Shir L’Yom Hashabbat,” which was written by Adam HaRIshon. It can be read that Adam is praising one who sings to Hashem on Shabbat as a way to achieve Teshuva. He quotes Rav Haim Vital that this is how Adam HaRishon himself was saved.
He relates that his father would sing the Zemirot with tears in his eyes, as though it was the first time he ever sang the words, offering explanations of the song. He would sing the Zemirot of Rabbenu Yehuda Halevi, Rabbi Shlomo ibn Gevirol, Rabbi Abraham ibn Ezra and Rabbi Yisrael Najara.
Thus, we see how it important it is to take the Pizmon book on Shabbat and sing at least a few Pizmonim with the kids. Not only is it an easy Misva L’chvod Shabbat, but also a powerful Tikun for Adam HaRishon and the person himself.