For starters, there is a fantastic library. Just remember to bring your Laptop with you in order to reap the benefits of Free WiFi. A trendy, social oasis in a slice of Sydney's history. A popular place to study, work, eat and read magazines. Read the current issue of some of the most popular magazines in the reading area downstairs.
Cafe Sydney is definitely a night time highlight for Customs House, and if you are in the area during the time that the Vivid exhibition is happening you are always treated to amazing light shows. During the 2013 Vivid there was a whole Dr. Who special which was incredible. They also use the space outside for artworks and displays (for example at the moment there is a gorgeous Rhino sculpture there advertising the Rhino exhibit at Dubbo zoo). All in all this is an area to visit whenever you can as it's always changing or something is always going on.
I love flicking through the magazines, and the library is a social hot spot. Can't complain the free WiFi - I love it. Sit on the couch with my laptop and plug away :) They say it is only for research purposes, and this is exactly what I am doing - working. A fun home away from home.
I don't know if it is an activity as such but Cafe Sydney is a must visit. I just love that view (pictured), and the food's not bad either) ! I recently saw an exhibition on level 2 there called "Capturing the Creation of the Sydney Opera House". The exhibits they have on there are often free and concerned with the history of this great city.