Please, suggestions on my 30 day vacation in Italy this October

I'll confine my comments to: You need to spend more time in Sicily. We've spent a total of five weeks there and ave not seen it all! If you're intending to go that far, please stay at least a week.

Based on my experiences during my naval career (3 years in Naples) my suggestions for that part of your trip would be Herculaneum (more interesting than Pompeii) and Ischia over Capri (personal preference.) And southern Italians are more forgiving (even if they are amused) if you take a brave stab at speaking Italian, even if your pronunciation is terrible. Don't miss the market days in Naples; they are a cultural delight.

Save Sicilia for another trip, 2-3 nights is not enough time.

Note that Rick Steves' books do not cover most of Italy. There is a rumor that a Sicily guidebook is in the works.

Will 1 full day do it, or will I need 2 full days in Pisa?

We spent about 2 hours in Pisa and that was enough for us.

Many people seem to just leave the train, see the leaning tower, and jump back on the train again. You may have better reasons to stay longer.

We spent an entire day in Pompeii. I'll remember to visit Herculaneum next time we are in the neighborhood.

You need about 30 minutes in Pisa. That's it - biggest tourist trap in all of Europe. Move on immediately and stay in Lucca for a few days.

The Shroud of Turin is not on display to the public at this time. There is an excellent museum at the Cathedral which tells the story of the Shroud.

Concur with Pisa comments. 1/2 day, at the most, is enough. Lucca is close and a favorite town of mine. Siena is another great city.

First of all, you should delete your email address from your post. You can do that by using the edit function.

Pisa is an easy 1/2 day trip from Florence by train. I'm told you can add Lucca for a full day out, but I haven't got there yet.
October may be too late in the year for the Dolomites. You can't just "drop down" to the French riviera. Look at the train schedules on bahn.com! You say 2-3 days in Sicily. The train is 9+ hours from Naples to Palermo. Are you planning to take the night trains? Sharing a 3-bed compartment? That may work if you are a sound sleeper. If you want to see Pompeii and Herculaneum, you might as well stay in Naples.

If you are on a budget, you'll want to buy train tickets on high-speed trains well in advance to get the deep discounts. As an example, the full price for the 3-hour train ride from Florence to Turin is €70, but a non-refundable, non-changeable ticket is €24.90.

I agree with the others--Sicily is not remotely a 2- or 3-day destination. It's awkward to reach, so don't go there if you aren't able to give it adequate time.

The French Riviera and Monaco also need more time than you can afford. Leave them for another trip.

Take a look at a guidebook that covers a lot of the places Rick doesn't have room for. There are so many other interesting spots in mainland Italy that you can reach in a just a few hours from one of your planned stops. For example, from Vicenza you can go to Padua.

Unlike some, I think the Leaning Tower is well worth seeing. However, the duomo and baptistery are just a few steps away; and that's it. Nothing much more that's compelling in Pisa. Half a day should do it.

Since you can't see the shroud, you might consider taking the Turin day, the Riviera days and the Pisa days and adding them to Sicily. That would give you a reasonable time there.(Just a note that we have been to Sicily twice, for a total of 16 days, and there's plenty we haven't seen.). If you do go, I suggest you fly from Naples.

If you're going to Florence, I would definitely stop at Siena, also. Be sure to see the beautiful cathedral and side library & baptistery.

I concur with others that Pisa is an over-touristy 1-hour stop.

We really enjoyed Turin - lots of interesting architecture. Yes, the Shroud is not on display. You will see the wooden box containing the Shroud. If keeping Turin, be sure to take the cog-train up to see the Superga & views, also.

I would suggest researching any Fall Oct. festivals in Italy. We've attended some, and they really make the town & people memorable. Enjoy your trip!

Piling on for Turin:
Shroud is only displayed on a very limited set of holy days. A replica/"photocopy" is on display. Turin a great Egyptian museum, with a collection second to the museum in Cairo. The Basilica di Superga is worth a visit. As is much else there. And it is a beautiful city.

I could suggest skipping entirely if viewing the Shroud is the only thing that matters to you. If it is, I might suggest instead Padua's St. Anthony Basilica or perhaps Assisi for your visitations rather than Turin. Or a day in Milan for the Duomo and the Last Supper. There's a ton of great Catholic sites in Italy, that might be better experiences than the Shroud. I couldn't really rate relative importance for you, not sharing the faith.

Wow, I am envious that you can spend a month in Italy! I have just a few thoughts on some of your questions.
Florence - give it at least the 4 days you're considering, 5 if possible. There is just so much to see.
Pompeii - absolutely go there. I suggest staying in Sorrento rather than Naples. Sorrento is beautiful, easy to walk around and just much calmer than Naples. It's easy to get to Pompeii from Sorrento on the local train. Do make time to go to the National Archaeological Museum in Naples, though. Many of the artifacts from Pompeii are there.

Assuming you take the advice others gave about skipping Sicily this time around, after you spend your time in Rome at the beginning of your trip, consider going to Naples/Sorrento/Pompeii area after that (4 days). Then take the fast train from Naples to Venice. Work your way back from Venice through the areas in Tuscany you want to see (including Florence). Very smart idea to spend the last few days resting in Rome before heading home. You're sure to have a few places left in Rome that that you'd want to see. Enjoy!