What is the ruling on going to the first mosque after the prayer has already been established there, while there is another mosque where the prayer has not yet begun, and is it better to go to the second mosque?

It is better to pray in the second mosque if you can catch the opening Takbeer (Takbeerat al-Ihram), based on the Prophet's (peace be upon him) saying: "Whoever prays to Allah for forty days in congregation, catching the first Takbeer, two absolutions will be written for him: absolution from Hellfire and absolution from hypocrisy." There is no harm in praying with the first mosque even if you catch less than one unit (rak'ah), for by doing so, you will attain the reward of congregational prayer.