I just got back from 3 weeks in Sardinia last night.
Roberto is correct about the airports and flights. You can not fly direct from North America to Sardinia. But there are dozens of ways to connect from various airports around Europe, and exactly where you fly through is up to you - there are probably a hundred possible options. We flew from Seattle to Amsterdam, then we spent a couple days there, then we caught a low cost flight from AMS to OLB (on Transavia), which was fine. Simple, not expensive. Could you spend a little more or a little less by flying via somewhere else? Probably. Lots and lots of choices so you will need to do the research on what makes the most sense for you. Personally, I always believe that if you shop for flights and the only criteria you use is the lowest possible cost, then you deserve the experience you get, but that's me (and then everyone complains how awful flying has become...).
Getting to the island by foot-ferry would surely be cheaper than flying, but would probably waste most of a day. Which commodity is more limited for you: time or money? It was a no-brainer for me.
Roberto is correct about the 3 airports there. Note that the airports each have different connections - OLB had many more flights/options than either CAG or AHO. OLB is the gateway to the "Costa Smeralda" which is the major tourist draw so more flights. My research showed most seasonal flights ended abruptly on 9/30.
Forget taking a car from Sardinia to Corsica (or vice versa) and leaving it there - that would cost you more than your flight from JFK.
You will need a car to get around on Sardinia. Driving was generally easy though occasionally challenging (it's Italy, 'nuff said). Be very careful about ZTLs.
We rented a car (Europcar-Italy) and their contract DID say ferry travel was NOT allowed; we planned to take ferries to some smaller islands off the coast of Sardinia so we asked them to clarify the policy. They came back and said ferries within Italy were fine, but not international ferries.
I'm sure that Corsica is also beautiful, but we had 3 weeks, and that wasn't enough to really see all of Sardinia that we wanted, and I drove a lot - probably more than many people would think reasonable (I could easily have spent a month there, probably longer), so we decided to save Corsica for a separate trip, and I'm really glad we did. There's plenty to see and do on Sardinia.
Roberto likes the northern half of the island. Yes, the northern half was great, but we thought the southern half was equally great. We loved all of it, and I'm really glad we went everywhere we did (short version: we flew into OLB, grabbed the car, and then drove all the way around Sardinia, counter-clockwise, mostly along the coasts, returning back to OLB to fly out). I'll just say it was spectacular, awesome, and one of our best trips ever (we're pretty seasoned world travelers so that's a high bar).
September is a good time to go, we had very nice weather overall, just a couple cool and windy days when we first arrived, and a very rainy last day, otherwise generally warm/hot and sunny. Went swimming several days (including last week) - towards the end of the trip is was starting to get a little cool but still fine for swimming when the sun was out. I came home with a sunburn (now peeling...).