Given the infrequent ferry connections between the Tuscan islands and the lack of facilities on all but a few, it makes the most sense, unless you have a week or more to devote to the archipelago, to pick just one island and stay put, perhaps making a day trip to one of the lesser islands. Elba has a ton of accommodations options in all categories, and Giglio is also fairly well equipped with hotels, B&Bs, and vacation rentals; Capraia and Giannutri have much more limited lodging. For travel in high season, always book well in advance. Visits to Pianosa, Gorgona, and Montecristo are restricted to day trips with guided tours and permits only; they have no overnight facilities. Elba and Giglio are the only Tuscan islands where you can bring a car.
Elba is a delight to visit off season, when you can trade the beaches for cozy trattorie and cool mountain hikes, and the scenery is no less stunning. A popular holiday weekend getaway for Romans and southern Tuscans, the Argentario also gets its share of winter visitors. The rest of the islands are all but deserted from November to March; even in April, May, and October, they're pretty empty, and you'll be hard pressed to find accommodations that aren't closed for the "winter."
As for the reliably warm summer months, June and September are the golden periods for a vacation, as with so many Italian islands. You'll have plenty of sun, the water is warm enough for swimming (though much more so in Sept), and hotels aren't charging their highest rates. Altissima stagione (highest season) is July and August (especially the last 2 weeks of Aug), when the islands are at their busiest and most expensive. For the smaller islands (Giglio, Capraia, and Giannutri) with fewer accommodations options, it's a must to book your travel well in advance.