Is it necessary to pray the whole 30 days taraweeh from one masjid because the Imam might complete the Quran? There is a an English speech/kutba at another Masjid and I really want to attend that. So which will be better for me? Pray in original masjid and miss out on speech or attend the speech and pray taraweeh there because apparently even Islamic speech's are of great reward IN SHA ALLAH.
You are free to go wherever you want to pray, however it is recommended to complete the actual taraweeh prayer with the Imam, meaning don't leave the mosque until the Imam you were actually following has finished the prayer (each night of Ramadan).
Note if you didn'*t do so you won't commit any sin (as we are talking about an optional act), but may have missed rewards.
This means you can pray each night at a different mosque if you want! You could even pray in different mosques in the same night, as I used to do when I was younger ('Isha' in the first mosque, two or four rak'a in the next mosque and so on), but as said above that may mean that you will miss rewards, as you won't complete a taraweeh night with the same Imam.
Anything else is already explained in the links suggested by III-AK-III in the comment:
Is it obligatory to pray Taraweeh in the mosque?
Is it necessary to complete recitation of entire Qur'an in taraweeh?