What should I do, and is expiation obligatory upon me, after I vowed to go out for ten days with Jama'at al-Tabligh if I succeeded in the exam, and I was blessed with success?

A vow of obedience must be fulfilled. Going out for da'wah (calling to Allah) is a great act of obedience. It is incorrect to describe going out with the Tablighi group as an innovation (bid'ah), unless it is proven that some of its members are truly involved in a real innovation. If you find a group among them adhering to the Sunnah, then the vow must be fulfilled, and you should go out with them, based on the Prophet's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." Expiation for an oath (kaffarat al-yameen) is not permissible except when unable to fulfill the vow.