7 Nights in Sicily

you will get lots of the same advice: Sicily is a 3 week vacation For only 7 days, pick 2 places to stay and make a few day trips if needed.

You might look at the travel time involved before deciding on an itinerary. Are flights booked? If you stick with these locations, see if you can fly into Palermo and out of Catania (or vice versa), or consider choosing locations that are closer together to maximize your time.

Sicily is likely to be very hot in July. Take a look at the actual, historical, day-by-day weather statistics on the website timeanddate.com so you know what you might be facing.

Personally, I'd concentrate on Palermo and Siracusa. A few hours in tourist-thronged Taormina was enough for me, but it's a beautiful place and some folks like it a lot. With seven nights (only six full days) I'm afraid you won't have time to see any of the Baroque villages outside Siracusa if you go to Taormina. There's a lot to see in Palermo, and a trip out to Monreale is a near-must; it's not far (accessible by public buses) and doesn't take a huge amount of time. Many folks also like to go to Cefalu, which calls for more time.

I would choose either the West or the East, not both. The island is large.
In the west, I would visit Palermo & Monreale, Segesta temple, and some beach time (Cefalu or the Scopello/Zingaro Reserve area). While visiting Palermo I would even stay in the Mondello district of Palermo. July is hot, and when it's hot, a beach is a good place to be.
In the East I would visit Siracusa and Taormina. I would also try to squeeze in some beach time by staying in Giardini di Naxos instead of Taormina. Giardini is just down the hill on the beach. If you are interested you can take a day excursion to the top of the Mt. Etna and see the volcano.

I don't think you have time to visit both the western and eastern side of the island. That would be too rushed, in my opinion.

I agree with acraven. There isn’t enough time for 3 places. If you stayed on the west, Roberto had some good ideas. Palermo and day trips to Segesta, Monreale, Erice/Trapani, Cefalù.

This is really helpful everyone. We will be flying in and out of Rome so flexible on the arrival/departure airport in Sicily.

If staying in Syracuse, should we stay in Ortigia island?

Definitely.