Naturally a huge variety so let us know the type of island and we can assist. Some are countries in their own right and some are part of but very independent to their sort of parent country.
The Balaeric Islands (Majorca, Ibiza and Mallorca are the biggest but there are others) are part of Spain. These islands are popular with Euopean families so avoiding school holidays from late June to late August is key. Each though have a lot to see and are much more than sun, sea and sand but that does appeal - for a mix of culture with a bit of warm weather then perfect in shoulder seasons.
Sardinia, Sicily and Corsica are larger islands - some visit them just on their own but some combine any of them with part of their associated countries as the sea and air connections are quite good. Personally I love the Aeolian Islands in northern Sicily - Volcano, Lipari and Stromboli are well worth a visit.
Malta and Gozo - partly British, partly Italian, partly Arabic and distinctively Maltese - again excellent for culture and relaxing in sunshine - no beaches really but plenty of really attractive rocky bays and lots to see and do.