time best spent in Siracusa

If you go to Ortigia from Taormina the remainder of the day (and the evening) should be good to enjoy Ortigia.
The archeological park will take a good portion of the day, but it depends on your interest in that. You might be done in a few hours and then go back to Ortigia which is a nice place to have dinner and stroll around. It’s a vibrant area with lots of night life.
Maybe you could try to do the park on the day of arrival, then enjoy Ortigia for the evening, then see Ortigia the next day and couple it with something else (like Noto)
You could visit Ragusa and even Noto on the same day trip, if you have a car. But maybe you can manage both with the trains too, although it might be a rushed visit and trains are probably not so frequent. Ragusa is the farthest but you should be able to get there in a bit over 2 hours. Noto is an intermediate stop along the way.
I don’t think you have much time for a beach, unless you give up Ragusa, and maybe visit Noto only, which is closer.
There are several nice beach locations in southwest Sicily, but to me the closest nice beach to Siracusa, which is also reachable by train, is Fontane Bianche, basically on the same train line to Noto.

I’ve only been there by car but the aforementioned towns are all served by train. Check the www.trenitalia.com website to see if the times work for you.

I would definitely forgo any hours in Catania. OTOH I had several days in Ortigia and enjoyed every minute. I didn't spend much time in the archaeological park, but I had already been to many other ancient Greek and Roman sights before I got to Ortigia.

We came from Taormina by train. We spent four nights in Ortigia. We arrived about noon and spent the rest of the day exploring Ortigia. I certainly would also skip Catania to have more time.

The next day we walked to the archeological park and museum. It was a full day. If I was to do it again, I would start with the museum as there were taxis outside the archeological park which we would have appreciated.

We took a boat ride our last day and had intended to take a bus to Noto but it was a gorgeous day and the forecast was for rain, rain, and more rain. So we just enjoyed being outdoors. If we had been more organized, we could have gone to a beach this day.

Chani hints at an important point. Your previous experiences will help determine how much time you spend at a location. We too didn’t go to the archaeological park for the same reason as Chani. On our trip we had a car and visited both Ortigia and Noto from the Catania area on the same day. For us, about 3 hours at each town was sufficient.

I can't imagine skipping Catania on a visit to Sicily.