How reliable are the ferries and is there one that would get me near Puglia? Very (well above Greek standards but below Swiss) and no.

Also how long would anyone suggest for mainland drive and any must-sees? Well, it would depend on your interests and dates. But if you managed to find only 5 days of things to do on Sicily then I would add about 7 days for Puglia and Basilicata. Calabria, I have no time for. Assuming you then drove up the west coast (you could go up the east coast) then I would set aside 6 days for the area around Naples, Rome needs 5 days and Tuscany/Umbria another 5 at your accelerated rate, though that assumes that you don't visit Florence as that requires 4 or 5 days on its own.

I suggest you get hold of the1000 page Rough Guide to Italy and start reading.

"is there one that would get me near Puglia"

No. You can ferry across the straights to the mainland then drive if you have a car, or perhaps easier would be to take a train from Sicily to Bari (the train will board it's own ferry) & pick up a car there. Or you can take a longer ferry to Salerno.

If time matters to you then you could fly from Catania or Palermo to Bari on Volotea airline

There is much more to see in Sicily than in Apulia. In 5 days you will not visit more than 25 per cent of all the gems of Sicily.
For 25 per cent of all the must see places of Apulia, I would count about 1-2 days.

It will take you about 5 or 6 hours to get to Puglia once you get back on the mainland from Sicily. Usually, the flight options are cheaper. My trip report on Puglia will give you an idea of what you can see in a week. 7 days/6 nights in Puglia

If you don't have a lot of time, maybe consider visiting Tropea and Matera on your way to Tuscany.

Based upon one trip only, DH and I took a train from Siracusa to Sorrento. We crossed the Straits of Messina and it was smooth. In fact, we were waiting for someone to tell us to de-train until we explored and found out the train was in the belly of the ferry. Cars were in the second level and people were atop enjoying a gorgeous day.

I don't have other 1st-hand info on your subject. Have a great trip.

Our first trip to Sicily 10 or more years ago was without a car (although we hired a private driver to take us to a hill town where my grandfather lived and some other sites). But we took rail from Palermo to Cefalu; bus from Palermo to Monreale. Train to Taormina. Train to Siracusa/Ortigia. Our next trip was with a car, based in Trapani, from where we visited three Greek temple sites, took a cable car to Erice. We have returned several times and now always have a car. This fall we will focus on the Baroque towns of the south again with a car.

Again, based upon one stay, we bussed from Palermo to Mondello and Agrigento and would have to Casale but couldn't match our times with bus schedule. We met a wonderful cab driver and did a deal for Monreale and driving all around Palermo. SO I would suggest using cabs and local transit in the cities. We bussed from Palermo to Siracusa and I thought traffic looked very reasonable. Friends then visited and said driving in the countryside was no problem at all. Once in Siracusa, we walked but did meet another cabbie who would have taken us on some local trips. We cabbed from the coliseum to Ortygia as it was a bit too much for us.

We drove all over the island. We prefer the freedom of a car, wherever we travel.

Seeing the highlights of Sicily generally takes 2.5 to 3 weeks.

Many people prefer to have a car; I prefer public transportation when possible -- its largely a matter of personal choice. There are a few places in Sicily that are difficult to see in an efficient way without a car, so I rented one for 3 days and had no trouble for the rest of my trip. There are trip reports here on Fodor's by people who have visited Sicily without a car at all -- look for the reports by GAC, Marija, and thursdaysd.

If you haven't already seen it, I would strongly encourage you to consult the Rough Guide to Sicily.

That could work, but depending on your interests, it might not be your best plan. Some things to consider:
- Taormina is beautiful, but can be so crowded with daytrippers during the day as to be truly unpleasant.
- Siracusa itself easily merits 2 days; stay in Ortigia.
- If you want to see one (or more) of the Baroque Towns of the Noto Valley, Noto is an easy day trip from Siracusa, but you would then need extra time in Siracusa. Otherwise, you could move on to Ragusa for a night or two.
- Palermo includes a wealth of treasures, easily worth 3 or more full days. Don't miss Monreale (plan on a half-day; its a short us ride from Palermo)
- No plan to see the Villa Romana del Casale? Or Segesta? Selinunte?
- "Going to Mt. Etna" can mean going to the top, which requires a full day and can be done by joining a tour from Taormina or Catania; your hotel will have information. It can also mean touring the slopes and some of their wineries; you can do that on your own and in a half day.

Skip the 3 nights at Catania in favour of places where is something to see like the Baroque cities or Casale or the other places mentined by kja.

Do you need SatNav?

Well, whether you can get by without GPS depends on your own map reading and navigation skills & comfort level.
For me, I am very comfortable with just a map, but like to always have the convenience of gps to tell me where I actually am.

For traveling outside the US I use google maps on my smart phone. I download local area maps while on wifi for offline use (you can do this for all of Sicily before you leave home if you wish). You don't need a data plan to then use this as you own free SatNav system. You can turn off your phone's cell data. Google maps will still give you turn by turn directions, but it won't have real time traffic updates without a data plan.

From my home computer I save numerous points of interest on google maps, then I can see them on my phone with no additional work.

Give it a try at home to see how it works for you.

You said you plan on only 1 or 2 nights in Palermo, which can easily take 3 or 4 days just to see the highlights. I'm not sure how you plan to fit Segesta in, but maybe it will work for you.

There are any number of blogs and online travel itineraries from people who enjoyed, 24, 36 and 48 hours in Palermo. A simple google search will yield those. Take a look and decide if any of these itineraries will meet your travel style and interests. We were able to see and do a lot our first time in Palermo, which was a two-night stay. Cefalu would be a nice spot just to relax.