Sicily for a month

You are not going to see Sicily from one location only, especially if you rely on the less than ideal public transportation in much of Sicily outside the big metro areas (Palermo and Catania). The island is large. Please use the www.Rome2rio.com to test some examples.

One decent base for some of your stay could be Giardini Naxos /Taormina.
Another one could be Palermo. If you prefer outside of big cities, and on the sea, maybe Mondello could do, but then you need to go to central Palermo for transport.
Siracusa could be another place for a few days.

Please heed Roberto's comments. It would be a terrible shame to tie yourself to a single base for your first trip to Sicily. There are so many wonderful things to see, but they are scattered all over the island. Just one example: Siracusa to Palermo is 5 to 7 hours by train, one way, and those are not the two most distant destinations you'd likely be interested in.

I agree that you should find at least two bases so you can make the most of your time...maybe one on the west part of island and one on the east. I was thinking of Giardini Naxos as well for one of the bases (on the east side). Maybe Trapani (on the west) for the other? Both are sea-accessible. If you have these two bases, then it makes most sense to fly open-jaw into Catania and out of Palermo, or vice versa. Also, travel by bus in Sicily is much more efficient between some destinations than by train (e.g. Siracusa to Palermo is several hours less by bus).

Pick three bases: Palermo, Taomorina, and Siracusa. Then you may be able to see much of the island by day tour/bus tour. Maybe rent a car for a couple of days, too.

Palermo would be a bad choice for someone who doesn't want to stay in a large town...it's the largest most chaotic town in Sicily (not that I would skip seeing it at all). Trapani is close enough and more laid back...I don't know if the OP would consider it "too large". Taormina will definitely be a wallet breaker...Giardini Naxos is very close and more affordable.

This is a good free guide to Sicily, along with a road map:
http://www.bestofsicily.com/roadmap.htm

I'm not sure public transportation is good enough to base in Trapani and do day-trips to Palermo. I vaguely remember being on a bus for quite some time, and the train takes a very slow, U-shaped route. I know the bus schedule is somewhere online, but I can't find it at the moment.

All I know is that I had 4 bases for my 18-day trip to Sicily and needed at least one extra base.

Palermo is certainly not a small town, but a must see.
If you don't want to stay in a city, then Mondello or Addaura would be nice and not too far. By car, assuming no traffic (therefore outside commuting hours) you can get to the city center in about 20 min, by bus it depends on the Greek Gods in charge at that time, 40-45 min is a good ETA.
If you want to stay further out of Palermo, then a car would be good. Staying outside cities and relying on public transportation do not always go together.
5 or 6 bases for Sicily would be good. Any less and you def need a car.

Trapani is an excellent base for a week...with a car.

October 2017 will be my 3rd 3-week trip to Sicily.
Sicily in October is perfect weather, smaller crowds.
I fly SwissAir into Zurich (from SF) then a quick connecting flight into Catania.
Then the 4€ Red Airbus (I think that's the name) to my first AirBnB rental.
I book 3-4 nights in Catania.... and start the dream.
As I sip wine on my patio I think of where I want to go next, and book the next AirBnb.
Always 3-4 days each city.
Transport by bus - which I find to be very inexpensive and far more efficient and reliable than anything to be found in California. Never rent a car. Never! I travel with my RS backpack suitcase and RS daypack.
3-4 nites in Ortygia, Enna, Palermo, Erice, Noto area, Agrigento. (Not in that order.) Don't miss Mt Etna.
This year I'm thinking I'll go to Aeolian Islands, Stromboli hike, maybe Cefalu, maybe back to some of my previous favorites.
Embrace the adventure and you won't be disappointed.... Sicily is truly amazing.

Never rent a car. Never!

My take is that absolutes like these make no sense. There is virtually no traffic in between the cities in Sicily. Driving would not be stressful at all outside the major cities. The buses and trains are inexpensive, but they don't run frequently on Sundays.

I agree with Agnes. While we usually eschew cars, Sicily is a place where a car makes sense, especially for a group. I even drove (into) Palermo!

And I agree with Laurel and Agnes. Angelina has posted this before; I'm not sure what happened to her when she rented a car, but my friend and I had no particular problems. And I say that as someone who does not even drive, so I have a strong anti-car bias. Of course you don't want or need a car in Palermo; I've read the same about Catania, but have no personal experience. The rest of the island can work fine with or without a car, unless of course you want to see more out of the way areas with little or no bus service. Part of our trip was a wonderful agriturismo near Caltanisetta, which would not have worked without a car.

We had 10 nights in Sicily, with a car. We had four bases (Palermo, rural area near Caltanisetta, Siracusa, and Taormina), and this worked well even with a short trip. With a full month, I'd spend a week in each base city you choose. And without a car, you will need more transit time, so you will definitely not want to try to see the whole island from fewer than 3-4 bases.

If you want the full story of my trip, including lots of info about car rental, here's my trip report: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

We are planning our first trip to Sicily in November. I'm hearing the weather may not be great. Thoughts? We live in Florida so the beach isn't important. Just want to see where our grandparents lived, maybe some cooking classes. Wine tasting.

Weather is unpredictable. Go to wunderground.com, enter one of the Sicilian cities you hope to visit (or, separately, more than one, such as Palermo and Siracusa), choose historical and monthly. Look at the actual weather data for November 2016, November 2015, November 2014, etc. The hour-by-hour temperature graph will give you a good idea of the range of temperatures you may encounter.

It appears that it rains, on average, about one out of three days in November.