Sicily first route: east-only or full loop?
With ten days, eastern Sicily makes the stronger first trip: Catania for Etna and the Ionian coast, then Siracusa/Ortigia for Noto and the Baroque south-east. Palermo and Agrigento are worth adding when you want to see more of the island and are happy to trade some slower days for longer journeys and another hotel change.
Full answer
Ten days in Sicily is enough for a rich trip, but crossing the island several times quickly eats into it. Eastern Sicily fits particularly well into that amount of time: Catania sits below Etna, Taormina is farther up the Ionian coast, and Siracusa/Ortigia and the Val di Noto take you south-east. Palermo and Agrigento pull you towards the other side of the island.
The two-base trip
Start in Catania. It gives you a working Sicilian city, Etna nearby and Taormina on the same side of the island. Then move to Siracusa/Ortigia, where the waterfront and old streets are worth having after the day-trippers thin out, and use it for the Baroque south-east, including Noto.
That already gives you a varied first trip: a city under Etna, Taormina’s coast and Greek theatre, the layers of Greek history in Siracusa, evenings on Ortigia and the pale stone towns of the south-east. You are not missing the point of Sicily by staying east; you are giving these places enough time to feel different from one another.
When to go farther
Palermo changes the feel of the trip. Its markets, churches and street life are markedly different from Catania and are reason enough to cross the island if that contrast matters to you.
Agrigento deserves the same treatment. The Valley of the Temples is worth the journey when it is one of the experiences you particularly want, rather than somewhere added simply because it sits between east and west on the map.
Once both enter the trip, expect at least one long cross-island journey, another check-in and less room for weather, a slow lunch or changing your mind. You can see more of Sicily in ten days, but the days become noticeably busier.
Car or no car
A car makes rural detours and the south-east easier, but it does not shrink the island. Without one, keep things especially simple and check current rail or replacement-bus arrangements before locking hotels; works and seasonal changes can alter individual journeys.
For most first visits, Catania + Siracusa/Ortigia makes better use of ten days. Add Palermo or Agrigento when seeing them matters more than keeping the trip slow.