Finally Sicily!

Take a look at the itinerary of the RS tour and see where they go.
I was on one of the very first Sicily tours that did most all of Sicily and then up the boot and ended in Rome. Now the tours are much shorter.
Here is a good web site with a lot of info on Sicily
Best of Sicily

Sicily is agreat place to visit however it is easier done with a tour co.
I did just that June 2017. Public transit from city to city is not the best and car rentals not worth it with parking troubles and thefts.
Consider it , there are choices out there, fully guided is a must for sicily.
You will be amazed how historical rich the area is.
I had a wonderful time and usually do not take guided bus tours.

No, you cannot see Sicily from a single base. Just too large. And of course it doesn't have any express trains. You need to pick up a guide book dedicated to Sicily, read it, and prioritize. Then you can figure out whether you need two bases or three. In ten days, I would not want four different hotels.

One thing you can do to save a bit of time (assuming you decide to visit both the E and W) is fly into Catania and out of Palermo, or vice versa.

Have you decided whether you will rent a car for at least part of the trip? It would significantly increase what you have time to see. You're almost guaranteed to have hot weather (though probably not as bad as it will be in July/August). One tangential benefit of having a rental car (assuming it is air conditioned) is that you will have a reprieve from the heat when you drive from town to town. I, on the other hand, spent a lot of time outdoors, waiting on buses. I don't think the train stations had a/c, either.

I didn't visit any of the isolated Greek temples but still spent 18 days on the island, traveling by train and bus. I could definitely have used more time. You'll want every bit of time you can scrounge, and still you won't have time get to all the areas of interest. So focus on part of them and plan to return.

I'd group the areas as follows:

The northeast: Taormina (gorgeous but the most touristy place on the island; has a Greek theatre), Catania (nice historic district, interesting market, but can be skipped), Mt. Etna (can be seen from a distance, saving a lot of time).

The southeast: Siracusa and the interior baroque towns of Ragusa (**), Noto, Modica, and Scicli. There's a Greek theatre in Siracusa, in a nice archaeological park. Also a good arch. museum. Ortygia, the historic area of Siracusa, is my favorite spot on the island. This is an area that I would not skip, but your priorities may be different.

Central Sicily: Piazza Armerina (pretty town with a nearby Roman villa with great mosaics--reachable by bus on some days of the week with careful planning, else taxi), Morgantina (Greek site that I haven't seen; reportedly under-touristed), Caltagirone (historic ceramic-producing town), Enna. The Sicilian interior is pretty, green and hilly. I think it's important to at least traverse this area rather than spending all your time on the coast. There is a rail line between Catania and Palermo that would at least give you a look at some of the countryside.

Southern coast: Agrigento (Greek site)

NW Sicily: Trapani (pretty and colorful) and the evocative hilltop medieval town of Erice. Also Greek sites at Segesta and Selinunte.

Egadi Islands: Accessed from Trapani. I was underwhelmed and consider them worthwhile only for people looking for uncrowded beaches. Otherwise, spending a whole day there isn't something I'd do unless I had more than 3 weeks in Sicily.

Palermo and surroundings: Monreale (magnificent basilica not far from city, accessible via city bus; arguably Sicily's top sight). Palermo's traffic is wild (do.not.drive.there) but there are lots of very worthwhile things to see: two great street markets (food), archaeological museum, many fabulous churches and private chapels. On the coast to the east is the former fishing village, now touristy beach town. of Cefalu. Worth seeing, but not as lovely or as dramatically situated as Taormina. With limited time, I think it is skippable, but it doesn't take long to get there from Palermo.

Aeolian Islands: Haven't seen these, but I think they're more interesting than the Egadis. However, with only 10 days, I would postpone them till the next trip. It just takes so long to get to and from islands. The ferry port isn't near a city you'd be very likely to visit otherwise.

I don't think you'll need to go looking for a beach. There will be water encounters at various points during your trip.

I don't think you need a tour or an RS book to easily get around. I used Lonely Planet and National Geographic and a book solely dedicated to Sicilian dishes, and it worked out great. I used buses and trains; except for Sundays, everything was straightforward. It does help if you speak some Italian as many locals don't. Acraven's post is excellent - I can't improve upon it....just get a guidebook from the library and start reading. If I were to go back there, I probably wouldn't hesitate to rent a car as there was no traffic in between the large towns and it would allow me to hit some spots that I didn't the first time. But you can easily organize a trip around your constraints. Not having a car just requires a bit more patience and flexibility; the upside is it's pretty relaxing just staring out the window (and there's beautiful scenery to enjoy).

I completely disagree with JS: You don't have to take a tour to see Sicily. I'm not the only one on the forum who has traveled independently there. I certainly agree that a tour with dedicated transportation will move you around the island much more expeditiously than you can mange on buses and trains yourself. However, I saw nothing to suggest that driving was particularly difficult outside Palermo and perhaps Catania (though the latter doesn't approach the insanity of Palermo). It's always a trade-off: Tours tend to visit multiple Greek sites like Agrigento and Segesta. Those are not my priorities.

Since your time is so limited, it is worth considering a tour, thinking of it as a sampler that will leave you wanting more, requiring a return. I usually don't like that approach to travel, because when you hop around, you are sort of forced to make another trip covering the same basic ground, filling in the gaps. But distances on Sicily are short enough that returning later to see Trapani/Erice and the Aeolian Islands (just as examples) would not require excessive travel time. That's very different from making a trip to Madrid-Rome-Athens, only to discover that you need to replow the same ground in order to see Barcelona-Florence-Venice-Greek islands. The fact is that 10 days will not be enough, no matter how you manage your time.

To get a decent overview you will have to move Hotels at least a couple times. It’s a big island. I just did it independently, but with a car. I’m sure it could be done by public transit, but would take longer.
Here’s what I did. These are what I thought were the most appealing sights to me and also worked logistically.

Fly into Catania.

Get car, drive to Siracusa for several nights. Day trip to Baroque towns (Ragusa, Noto). I also day triped to Mt Etna but this would be easier from Catania or Taormina. I was limiting hotel stops this trip.

(This is where I wanted to go to Agrigento for a night to see the temples but was too squeezed on time)

Drive to Scopello - beach time and day trips to Segesta and Erice.

Drive to Palermo, drop car, stay several nights then fly out. From Palermo it is easy to get to Monreale by bus - that’s barely a day trip at 30 minutes away.

I’m heading there in the fall for about 3 weeks, but will be driving rather than depending on public transportation. The main sites we intend on seeing, although the itinerary isn’t set in stone, Erice, Segesta, Monreale, Selinunte, Agrigento, Villa romana Casale, Mt. Etna, Taormina, and Palermo. I also plan to visit a number of smaller sites between the main sites. Even with 3 weeks we won’t be able to see everything we’d like to.

We have spent a total of five weeks in Sicily over two trips, all on trains and buses, and haven't yet seen it all. If you want to stick to public transportation, you might choose one side of the island (east or west) for this trip and plan to return for a second trip to see the other side. Taormina (if I go back there, I'll stay down by the beach) and Siracusa (stay on Ortygia) were two of our favorites, though we enjoyed every single place we visited in Sicily. You'll have a wonderful time. It will be HOT!

Janet's post reminded me: Because of the heat, it will be advantageous to select hotels that are as convenient as possible to sightseeing attractions so you can retreat to your room to escape the midday heat if necessary. That may not be practical in Palermo since sights there are fairly scattered and you'll probably want a place close to parking, but it's a reasonable goal if you stay in smaller cities and towns.

I said above that I think you need to move around to different cities and different hotels in Sicily. I stand by that, but just in case that's not possible...I think if one was going to go to Sicily and wanted to stay in only one place I would suggest either Siracusa or Palermo.

Palermo is a fascinating city with layers of history, and you can day trip to Cefalu, Segesta, the beach, and Trapani/Erice. I would NOT drive here - you can use transit or day tour groups.

Siracusa is smaller, more charming - staying on the island of Ortygia is beautiful and gives the water views you mentioned. It also has some great Ancient Greek sights in town. You can also day trip to Mt. Etna, the Baroque towns, and Catania. And I assume beaches. You can have a car here - easier to drive in than Palermo. I assume you can use public transit as well.

If you want to see what I did with 10 nights in Sicily, visiting with a friend and driving, here's my VERY detailed report from April-May 2014: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

We had 4 different accomodations over that time, and it wasn't a problem. We drove ourselves, and it wasn't a problem.

"I do enjoy water views especially in the evening after a full day of exploring."

Our Palermo B&B had a water view from the terrace!

I spend 6 months in Sicily and it was not enough, I love it! Rent a car, its a lot easier although they now open a train from Catania that facilitate transport but still easier with a car.

I would split your trip in to Catania/Palermo.

Catania as your home base to visit; Siracusa-Ortigia, Noto, Ragusa, Agrigento (Valle del Templi, best kept Greek ruins in the world), Piazza Armeria (Via Romana del Casale), all those are day trips. In your way to Mt. Etna you can stop at Aci Reale and /or Aci Trezza for apperitive, and then do some wine tasting at The Gambino Winery. Last but not least, Taormina to see the Greek-Roman Amphitheater who still in use today. Short ride from Taormina is Forza d'Agro (old village frozen in time) to see the church from The Godfather and have some gelato at Lantohiani (spealling??) or skydive. Back in Catania go see the fish market (il mercato di pesce) in the morning then at "sera" do the "passegiata" and later enjoy the food, specially il fruti di mare (seafood). It is awesome. The bus system from Catania to Taormina is very good and realiable, it drops you near the centro storico of Taormina (5 min short walk).
Palermo as your home base; go to near by Segesta (another great Greek temple or ruins), then head to Trapani, Erice, San Vito lo Capo, the Aeonian Islands....