Is it permissible for my wife to shake hands with my father?

Yes, it is permissible for the husband's father to shake hands with his son's wife, because the husband's father and his son (from another woman), and the wife's mother and her daughter (from another man), all become mahram (unmarriageable kin) through affinity (marriage). The husband's father is forbidden to his son's wife, and the husband's son is forbidden to his father's wife, and the wife's mother is forbidden to her daughter's husband immediately upon the marriage contract. However, the wife's daughter is not forbidden to her mother's husband unless he has consummated the marriage with her mother. Therefore, the husband's father is mahram to his son's wife, and it is permissible for him to shake her hand, be alone with her, and travel with her.