Is it permissible for a man to shake hands with his wife's mother (mother-in-law) and travel with her?
Answer:
Yes, there is no harm in doing so because she is among his Mahram women. Undoubtedly, Allah (Glorified and Exalted) has permanently prohibited a man from marrying his wife's mother (mother-in-law). Hence, she is among his Mahram women, which means:
There is no harm in shaking hands with her.
There is no harm in traveling with her since he is her Mahram.
However, if there is a fear of fitnah (temptation) or the arousal of desire, then he should avoid both shaking hands and traveling with her. If there is no concern of any prohibited act or temptation, then it remains permissible for him to shake hands with her and travel with her.
Since she becomes his Mahram due to his marriage to her daughter, this ruling applies, as Allah (SWT) states in the Qur'an:
"And (forbidden to you in marriage are) the mothers of your wives." (Surah An-Nisa: 23)
(Fatwa by Sheikh Saleh bin Fawzan bin Abdullah, may Allah preserve him)