SICILY:
Well, since you intend to visit limited places it is up to you. However the logistics to get to the Aeolian islands are such that you need to spend at least 4 nights in those islands. If you want to add Siracusa, maybe another 2 or 3 nights in Siracusa. The most logical airport in this case would be Catania. You might need a night or two in Catania after arrival and/or before departure, depending on the flight times. Going from Catania to the Aeolians might take the best part of the day (ferries are in Milazzo). Same story coming back. You do the math on how many total nights you need.
SARDINIA:
I’m not sure why someone wrote that Sardinia is much smaller than Sicily but it is not the case. Sardinia is nearly as big as Sicily (9,301 sq.mi. vs. 9,907sq.mi.). Since Sardinia has a less developed transportation infrastructure (both roads and rail), traveling in Sardinia will be more time consuming. A rental car is therefore a must. Sardinia does not have the richness of historical sights that Sicily has, but the natural beauty of the island is stunning, especially the sea, which is a scuba diver’s paradise. If you are interested in natural beauty you need at the very least a full week to see a small portion of it, for example the Maddalena Archipelago and the Gallura in the north (since you want to see Corsica). If you want to see the whole island then you need a lot longer because highways are mostly two lanes and curvy therefore it takes forever to go from place to place.
CORSICA:
Corsica is also a natural beauty. Just more lush and rugged than Sardinia. Same comments I made for Sardinia also apply here. You also need extra money because prices are at least 50% higher than Sardinia. 5 nights is the minimum I would plan to make it worthwhile, it you can keep busy for weeks if you want.
There are no direct links between Sicily and Sardinia therefore you’ll need a layover in the mainland (likely Rome).
There are also no direct air links between Sardinia and Corsica, however there is a car ferry that can shuttle you back and forth across the Bonifacio strait between Santa Teresa and Bonifacio. The 12 km ferry crossing takes about 45 min.
I don’t think there are non stop flights from Corsica to Italy, therefore plan to take the ferry back to Sardinia. Olbia is the best airport for northern Sardinia.