Sicily and Sardinia

Strongly consider choosing one island. There is so much to see and do on both--it seems like a lot of time, and it is provided you don't insert the island change in the middle.
I would get island-specific books and read up on the public transport before deciding. I think Sicily will be much easier than Sardinia in that regard. I would not recommend Sardinia by public transport unless it is a very targeted, well chosen (limited) plan.
I flew Lufthansa via Munich, very convenient.

Oof, I wouldn't try Sardegna without a car.....it's very neat, but I can't imagine what I would have been limited to if I didn't have one....

Yeah, listen to the wise advice in the posts above.

3 weeks is a good amount of time for Sardinia. Or for Sicily. But not for both. They are each quite big, and it takes some time to get around. There are great places on both, but they're inconveniently scattered all around. You really do need a car for Sardinia, as public transport there is pretty thin.

In October I'd spend the whole three weeks in Sicily where there is so much to see.... In Sardinia the shorter days in October and possibly poor weather might not be the best use of your time

We are planning a 19 or 20 day trip to Sardinia and Sicily in September/October 2026. Here is my tentative itinerary.
Fly into Olbia, rent car, stay somewhere near the port for boat tour of Maddelena Archipelago which we will do on our second day there. Stay 3 nights
Drive to Alghero and stay 2 nights, visit town and beaches
Drive to Bosa, stay one night.
Drive to Cagliari area, stay 3 nights. Visit town and beaches.
Drop off car at Cagliari airport and fly to Sicily.
Fly into Palermo and stay 2 or 3 nights, possible day trip by train to Cefalu
Train to Taormina, stay 2 nights
Train to Syracuse, rent car, stay 3 nights. Road trip to Ragusa and Noto.
Return rental car, train to Catania, stay one night and fly home.
Maybe a bit ambitious but I think it will work.

3 weeks is a good amount of time for Sardinia. Or for Sicily. But not for both.

Three weeks is a relative luxury for most travelers - so ten days for each doesn't seem unreasonable - it may be longer than a typical visit. It isn't required that one does an exhaustive circumnavigation around Sicily to have a rewarding visit. If the OP wants to do both, then perhaps concentrate on constructive suggestions.

I was planning to do both islands too but changed my mind. Way too much travel in such a short period. I think it would be exhausting for an adult much less a toddler. So much time in transportation.

Sardinia is famous for its breathtaking beaches and turquoise sea waters, but in October the weather may not be ideal for beach time, it depends. In any case I wouldn’t consider Sardinia without a rental car. Sicily has more to offer, in terms of historic sights, so spend more time in Sicily, unless snorkeling and scuba diving is the most important thing for you.

It is absolutely possible to visit both these islands in three weeks! Of course you won't see everything as they are quite large but I don't think you would expect to.

In Sicily, if your top sites are Taormina and Ortigia I would just stick to the east side of the island. You can fly into/out of Catania. I would definitely suggest a visit to Mount Etna as well- there are lots of tour companies that will drive you up as I imagine with a toddler you're not planning on a lot of hiking (or perhaps you are- either way it's worth visiting). The public transport in Sicily is pretty good between trains and buses, at least for the more popular areas we've been to.

In Sardinia, it is difficult to get around without a car. I would definitely suggest looking into renting one and then figuring out what part of the island you want to concentrate on. The north/Maddalena Islands have lovely beaches but as has been mentioned above October will be too cold for the beach. Good luck with your planning!

The east side of Sicily has decent public transportation. We flew into Catania and took a bus from the airport to Taormina were we stayed three nights. We did take a tour to Mt. Etna but I am not sure that would be appropriate with a toddler.

We then took a taxi to the train station and took the train to Siracusa where the owners of the place we stayed picked us up and took us to Ortigia. We had planned a bus trip to Noto but didn't end up going.

The train also goes to the Baroque villages of Ragusa and Modica. We rented a car though as we left Siracusa and stayed in Ragusa and visited Modica. The east and western sides of the island are more difficult to connect with public transportation so we rented a car.

In Sardina, I am not much help. I have only been to Cagliari which you certainly don't need a car for but that isn't the prettiest part of the island.