Is there a sin in moving to another mosque to perform the Tarawih prayer because its Imam is distinguished by good recitation, supplication, and lessons, and will we be rewarded for that, and will the one who transports us be rewarded?

If the mosque you attend is characterized by a large congregation, the imam's good recitation and voice, and the presence of beneficial lessons and supplications, then going to it is better due to the increase in reward that results from it. And whoever helps you go to it is considered a helper in righteousness and piety.