Okay, Lokrum Island is officially excellent. It's a literal hop, skip and a jump from Downtown Dubrovnik and offers everything one could desire for a lazy day of exploring—swimming, cliff jumping, ice cream, peacocks, a 19th-century fort, a 12th-century monastery, and a weird, inland lake. Oh, and it is also home to a Game of Thrones Museum where you can sit on the *actual* Iron Throne which, was left behind by the crew because it was ostensibly too large to lug elsewhere.
The island's shores are rocky but perfect for an afternoon of swimming, sunbathing or leaping off boulders into the refreshing Adriatic. Parks and picnic areas abound as well as an assortment of colorful locals—peacocks, bunnies and a healthy sprinkling of "free body culturists" (aka nudists).
BYO: Swimsuit, water shoes/socks, towel, sunscreen, water. Ferries depart hourly from Dubrovnik's Old Harbour. The ride lasts 15 minutes and tickets cost 40 HRK / €5 per adult.