This is a pretty little Island just few minutes by boat and a 100 kunas away from Dubrovnik. You can just ask around and people will direct you to the right boat that takes you there. I think it leaves on the hour or half an hour. It's a good half day trip. I spent about 5 hours there. There is a big map pointing the main attractions once you get there, so everything you want to see is clearly marked. Here Five things you could do: 

1. See the history
   The Benedictine abbey and monastery were founded in 1023. The last Benedictines left the island in 1808. Legend has it that on their last night, the monks put a curse on the island and anyone who tried to seek it for their own in the future. There were actually few more legends associated with the islands that you can read about when you get there. Also, it's been used to film some parts of the Game of Thrones. 
2. Enjoy the nature!
   Very close to the starting point is the botanical garden, which is nice to walk around. 
3. Chill with the rabbits and the peacocks
   They are just running wild around. They are mostly out when the weather is cooler in the morning and the afternoon. I got there at noon, so there weren't as many.  
4. Dip in the dead sea
   I was so tempted to taste it just to see if it was really salty, but I got a good proof when my copper bracelet turned green! Chemistry 101 there for you. It's located at the beginning of the island, but coevered with trees, so we were slightly confused on where it was. 
5. Dip in the empty ocean shores
   The weather was hot in September, but the water was on the cold side. Like it gives you shivers once you jump in! Note: When I was reading about this island online after the trip, I found out there is a part for nudes, so be aware of that! Although, when I was there everyone was at least wearing their swimsuit.

PS. Food is SO overpriced there, so pack something with you if you want to save up.