Hello IvanSusanin1613,
I wish I had more help. The only actual lead I may have is in Florence: https://www.museofiorentinopreistoria.it/en
And here is a list of potential Etruscan sites in and around Rome: https://www.romecabs.com/blog/docs/10-etruscan-museums-visit-etruscan-tours-rome/
As for other places in Italy:
Bologna was in a big Celtic area and the Archeological museum has a surprising number of Etruscan, Celtic and other previous culture stuff. The museum itself is interesting in that the first part is kind of what you expect but if you keep going deeper the later parts are clearly kind of jumbled up 50s feeling displays and obviously way more artifacts than they can handle. So there cases with just spoil piles of artifacts from this site and artifacts from that site.
The small but interesting museum in Verona has a surprising amount of pre-Roman Etruscan, Greek and some Celtic artifacts as well. Apparently the Etruscans were big Hercules fans and traded pretty heavily with the Greeks.
In Brescia there are great Roman ruins (nothing to rival the south, of course) but the actual mostly intact temple you get to tour is actually from a Celtic cult goddess that Rome made them close to fully be a Roman city but it was the same rank right beside the big Roman gods.
I haven't yet been to Grosseto but their The Museum of Maremma Archaeology and Art is supposed to have a great history spanning museum.
The area south of Salerno has Paestum which was Greek, the area has a lot of pre-Greek cultures meeting and getting along - or not - in this area. The museum at Paestum touches on this some and there are sites in the area as well Etruscan stuff in towns in nearby.
The only truly ancient sites I know of in Italy are in Sardina that has Neolithic gravesite and Bronze age standing tombs.
https://archaeology-travel.com/thematic-guides/nuragic-civilisation/
Pre-Roman Iron Age necropolis unearthed near Naples
https://www.thehistoryblog.com/archives/70135
Sorry to mostly point out where you aren't going but hopefully bumping this post will get you some more input.
Have a great trip,
=Tod