Montenero Alto is definitely worth a visit, perhaps using (with a normal city bus ticket) the nice funicular that leaves from Piazza delle Carrozze, in Montenero Basso.
The Sanctuary dedicated to the Madonna di Montenero, protector of Livorno and Patron Saint of Tuscany, is very interesting; the Church, in rich Baroque style, also houses an impressive collection of humble and rich ex-votos, the result of intense popular devotion over the centuries.
In the Famedio, the portico in front of the entrance to the Sanctuary, there are cenotaphs and tombs of people who distinguished themselves in the world of culture, politics, art... who were born or lived in Livorno.
Furthermore, from the Sanctuary square and its surroundings you can admire magnificent views from above not only of the city, but - proceeding from southwest to northeast - also of the sea and the islands of the Tuscan Archipelago (and on clear days even of Corsica), of the coast up to Liguria, of the Apuan Alps and of the western Apennines.
Finally, going up via della Porcigliana to Castellaccio, you can reach the top of the Montenero hill (over 300 m above sea level) and from here enjoy a wide, splendid panorama.