Click on my name to see my trip report from 2005. I assume a car is not in your plans.
I would stay in Taormina rather than Catania---you can still day trip to Etna from there and the town is so much nicer.

Here is my photo gallery:

http://www.slowphotos.com/photo/showgallery.php?cat=3828

You definitely have touched the highpoints of Sicily, AND I would urge you to spend a night or two in Taormina, only 40 min by train or bus north of Catania, on the coast, spectacular medieval town on a cliff with upscale shops, staggering views of the Med, Mt Etna and a nice cable car to the beach. Be sure to get some slices of that rectangular pizza from little shops and the (non alco) drink chiosco, from kiosks, a fruit flavor experience! Also they have a Greek Theatre there built 2500 yrs ago! If you are very fit you can walk up from the train, but I rec conserve your energy and take the cable car.
In Catania, be sure to go to the flea market/fish market and tour the cathedral.
Malta has an excellent bus system and you won't need a car, you will be across the island before you know it! Be sure to see the Christian crypts, the museum in Valetta and that ancient altar that was built so long ago, that no one knows what god it was made for, it's behind/in a luxury hotel, so look for it!

You might want to look at my trip report, even though it involves non-public transportation travel; just click on my name to find it.

I think that Selinunte is somewhat removed from Trapani for a day trip via public transportation.

I would like to add my vote to stay in Taormina rather than Catania. We stayed for 3 nights and found so much to do. The views are the most spectacular we found in Sicily. You are so lucky to be spending 3 weeks in Sicily - we had 10 days and loved every minute.

My wife and I did a group tour to Sicily and Malta last Fall. Your itinerary will hit all of the highlights of our tour and more. We found Sicily quite interesting with quite a few surprises. Malta we loved. It is quite easy to get around on the local buses. While there we made sure that we would get to enter the Hypogeum, a 3 storey underground temple (they think) carved out of the rock. You must secure your entrance tickets well in advance to ensure to can get in while you are there.

I posted a trip report of our tour. Within it I have provided a link for the purchase of the tickets for the Hypogeum.

Thanks so much for the great advice and all the wonderful trip reports. Looks like I better reconsider Taormina. DaveJJ -- special thanks for the info on Malta which is not easy to come by.

Thanks to all for the helpful input, Trip Reports etc. on this forum (and also to Vagabonda on Trip Advisor). Here is my second attempt.

Overnight Ferry from Naples to Palermo
Palermo (6 nights) -- includes day trips to Monreale, Cefalu, and Agrigento
Train to Trapani
Trapani (5 nights) -- includes day trips to Erice, Segesta, Selinunte, and Marsala
Train to Catania
Catania (3 nights) -- includes Bellini House, Mt. Etna and possibly Taormina
Bus to Syracuse
Syracuse/Ortiga (4 nights) -- includes day trips to Noto and Palazzolo Acreide
Bus to Ragusa
Ragusa (2 nights)
Ferry from Pozzallo to Malta
Valletta (3 nights) -- includes Hypogeum, and HoHo bus to Mdina etc.
Gozo (2 nights)
Valletta (1 night)

Given how much time you're able to spend in Sicily, I'd recommend that you give Taormina at least a day. Taormina is touristy but, for me, not tacky, and I never felt like tourist prey. It was the most physically beautiful of all the places we visited. Ragusa Ibla is lovely but has a somewhat Disney quality because most people live in the newer town. You can click on my name for my trip report.