Lorie, on my trip last year I was on Sardinia and Corsica for 11 full days. You’ll receive different opinions as to how long a stay is necessary to see these beautiful islands. It depends on how deeply you want to delve into them and how fast or slow you like to move. Easiest thing to do is to fly JFK >FCO>CAG. Rent a car in Càgliari, go up the east coast of Sardinia and take the 7 AM ferry across from Santa Theresa to Bonifacio. Once on Corsica, I drove north and stopped in Porto Vecchio and Corte on the way to Bastia where I spent two nights. I did a day trip around Cap Corse - amazing scenery. If you feel comfortable driving narrow, twisty roads, visit Barcaggio at the extreme northern tip of the island. I drove up the east side of the Cap and down the west side to Saint Florent and then took the D81 back to Bastia.
From Bastia I headed through the mountains on the D84 to the west coast through Evisa to the coast town of Porto - a marine sanctuary - again detouring from the main road to stop in the hill town of Ota by taking the D124 to Porto. After an overnight in Porto, I took the D81 to Ajaccio and the T40 back to Bonifacio.
Then it was the 5 PM ferry back to Sardinia where I went down the west coast to Càgliari.
I spent 2 nights in Bastia at the Best Western Bastia Center (meh!), 1 at Logis Hotel La Lonca in Porto (good hotel, free parking, beautiful view from my terrace, good breakfast), 1 at the Hôtel Résidence du Centre Nautique (quirky but nice, right on the Bonifacio marina, free parking, super convenient).
In Bastia I had very good moules et frites at Café Albert 1er on the Blvd Charles de Gaulle on the edge of the park and the ferry port and another great tangine at La Medina on Avenue Emil Sari. For authentic Corsican cuisine, try Restaurant La Moulin in Porto. Without exaggeration, there must be more than 50 restaurants around the marina in Bonifacio. I had a nice ravioli with langoustines at Coté Fruits de Mer.