Is it permissible to congratulate people on the advent of the month of Ramadan, or is that considered an innovation (bid'ah)?

There is no harm in congratulating one another on the advent of the month. The Prophet, may Allah’s prayers and peace be upon him, used to give glad tidings to his companions about the coming of the month of Ramadan (definition) and urge them to pay attention to it. Among these sayings is: "Ramadan, a blessed month, has come to you. Allah, the Mighty and Majestic, has enjoined its fasting upon you. In it, the gates of the heavens are opened, the gates of Hell are closed, and the rebellious devils are chained. In it is a night that is better than a thousand months. Whoever is deprived of its goodness is indeed deprived."