Spain and Sicily

You will definitely need to fly between Spain and Sicily. A train would take about 30 hours.

Honestly, with the flights you've already booked, I'd stick to Madrid (perhaps with a side-trip to Toledo) and Sicily. Even that gives you much less time than is ideal for Sicily (where I'd recommend at least 2 full weeks). Barcelona, which I love, is a sight-rich city where you'd benefit from having quite a bit of time. Because transportation to the Basque Country isn't speedy, it would take you the better part of three days to have just one day in San Sebastian/Bilbao--and each of those cities really needs a full day.

If you do a decent job on Madrid, Barcelona and the Basque Country, there will be almost no time left for Sicily. (Could you change your return flight and fly back from Barcelona, leaving Sicily for a later trip? It would be a better plan.)

According to Skyscanner, there are non-stop RyanAir flights from Madrid to Palermo, though I think not every day. Vueling has non-stops from Barcelona. There are also non-stops into the slightly busier Catania airport from both Madrid and Barcelona. Since you're flying out of Palermo, I'd recommend flying into Catania and traveling more or less toward Palermo so you don't have to make a full loop around the island.

For your proposed destinations in Spain, I'd stick to trains. In Sicily, a car might make your travel faster outside the major cities, depending on your itinerary. You shouldn't try to drive in Catania or, especially, Palermo. Whether the car is truly helpful elsewhere depends on where "elsewhere" is, which will be affected by how much time you have left after you visit Spain. I don't think you're going to end up with enough time for some of the harder-to-reach places, so train or bus service may be adequate.

October
3. Arrive Madrid (5)
4. Madrid
5. Day to Toledo
6. Madrid
7. Day to Segovia
8. Train to Barcelona (5)
9.10.11.12. Barcelona
13. Fly from Barcelona to Catania, then transfer to Taormina (2)
14. Taormina
15. To Siracusa (2)
16. Siracusa
17. Via Noto to Modica (1)
18. Via Ragusa to Agrigento (1)
19. To Palermo (3)
20. Palermo
21. Day to Erice or Cefalu
22. Depart Palermo

A car is probably the best way to get around Sicily.

Thank you for the responses.
David, this sounds like a great itinerary but do you think we would better off just spending our time in Spain? We've never been to Spain and never to Sicily. If we were to concentrate on just Spain what would you suggest?

Are you not booked to fly home from Palermo ?

We are booked to return from Palermo but it can be changed.

In that case, I heartily agree with sticking to Spain. Sicily can be a separate, wonderful trip.

That leaves you with enough time in Spain to see Barcelona (with a side-trip or two), Madrid (ditto) and either the Basque Country or Andalucía. If you want to do both of the latter, you'll have to be fairly brutal about time spent in the big cities, and you'll spend more time than I'd like, just moving back and forth. Granada and the Basque Country are both time-consuming to reach. Try hard to put Bilbao at the beginning or end of your trip so you can make it one of your flight gateways, avoiding the need to travel both into and out of the city on the ground (about 6-1/2 hours to Barcelona and Madrid).

Other Basque Country tips: Bus is the way to travel between Bilbao and San Sebastian. Bilbao has a larger medieval district than San Sebastian does; don't cut the city short. There are other interesting destinations in that part of Spain. If it develops that you'll have extra time available, we can suggest other places to see.

If you do decide to include Sicily , i recommend taking a guided tour.
I also am active and usually independent traveler however i took a gate one tour in June 2017. It was great, 9 days and fully guided and relaxing as well.

If you want a contrast and some variety then I see no problem with including Sicily. I think my breakdown of 5 nights in both Madrid and Barcelona, followed by 9 nights in Sicily, does justice to all destinations. This decision is all yours to make.