Some love it for original exhibitions and awesome concerts, other for possibility to have a delightful picnic or sunbathing at the very heart of the city and I love this place for that incredible feeling of peace that I gain every time chilling on the beach and looking at all those people rushing somewhere on the other side of the Neva river.
Have a look at the most controversial statue of Peter the Great, peek at the all-season sun lovers (the strongest ones come for a sunbath even in February), try to hit the hare by Ioannovsky bridge with a coin for good luck and hear traditional cannon shot at midday (on summer Saturdays you can also observe change of guards right after that).
The island is open from 6 am till 10 pm, all the museum parts close at approximately 5 pm, but you can still walk around the fortress.