Perugia is the capital of umbria, and I am now studying languages at Perugia University for Foreigners. Because people are in Italy, their daily communication is in Italian, and the communication between classmates is in Italian, so it is very practical. The teacher speaks grammar very well. It is suggested that there should be a new lesson every month. For details, you can check the monthly courses on www.unistrapg.it website, and there will be a simple entrance exam before entering the school.
Siena University for Foreigners is another authoritative foreign language institution with the same name as Perugia University for Foreigners. It is located in central Italy, near Florence. You can view the course at www.unistrasi.it
As for your idea of going to a big city, I don't object, but a big city does have its disadvantages. To learn this thing, you should calm down and study quietly. From this perspective, big cities are too tempting.
I recommend some for you, www.arcodidruso.com from Arco di Derousseau in Rome.
Dante alighieri Italian Center: www.clidante.it
Italian www.italiaidea.com.
Www.torredibabele.com Babel Italian Language and Culture Center
But these are private, and the fees will be higher, but the quality of teaching is still guaranteed.
If you want to choose an authoritative one, foreigners should choose Perugia University and foreigners should choose Siena University.