What is the correct action to take when dealing with the innovation (bid‘ah) of shaking hands after prayer and other innovations? And is it advisable to emigrate to a non-Islamic country to complete studies and then return to the Levant, or is it better to remain and be patient in the current country?

Persevering in shaking hands after the salam with the supplication "Taqabbal Allah" (May Allah accept) is an innovation (bid'ah). This can be explained gently and kindly to the one who shakes your hand. As for residing in a country of evil deeds or migrating to lands of disbelief, scholars in your country should be consulted regarding this.