Hi, Giangilberto from Milan, Sanremo is a place I visit every summer.
As the user Slate mentioned, the old town, known as "La Pigna", is Sanremo's historic medieval core, founded around the year 1000, it is well worth a visit. Situated in the upper part of the town, it features the characteristic narrow alleyways and beautiful viewpoints typical of Ligurian historic centers.
Another interesting spot is Forte di Santa Tecla, formerly a prison, it is now managed by the Ministry of Culture as a museum, where visitors can explore the site and view rotating exhibitions.
https://fortesantatecla.cultura.gov.it/
However, my top recommendation, if you have a car, is to visit Bussana Vecchia, a small village right next to Sanremo. Devastated by an earthquake in the early 20th century, the village was reborn in the 1960s and 70s thanks to people from all over the world, many with ties to hippie culture, who began repopulating and partially rebuilding the area, eventually settling there.
It is my favorite place to retreat to whenever I visit Sanremo.
If it is open, stop by Osteria degli Artisti for an excellent lunch or dinner.