Additional activities should be tackled between main story missions - each district of the city has a several dozen of such activities, which are divided into several groups (described below). Completing the activities allows you to earn secrets and to unlock new abilities, but it is also needed to weaken the D.U.P. influence in the area. More on this topic can be read in the "Step 9" section.
In the "Step 7" it was mentioned, that Blast Shards are used to acquire new powers and finding them is one of the additional activities in the game. The shards appear in two "forms", and it's easier to acquire them in their "passive form", because all you have to do is to get close to them and simply collect them. Some of the shards are harder to get - by destroying environmental obstacles hiding them from you or by shooting down Tracking Drones. In the case of the latter, you have to track him down (climbing up a roof will surely help), bring him down to the ground with your ranged attacks (Smoke Shot is best suited for it) and examine the wreckage afterwards.
Collecting as many of those shards as possible should be your priority, as each one of them guarantees you one skill point, which means, that the faster you collect them, the sooner you will be able to purchase all of the most important abilities.
Destroying Security Cameras is yet another available activity. You won't get any special reward for doing so, but you will be one step closer to assuming control over the given district and you won't have to be careful not to get seen by the camera.
Another activity available to you is the exposure of the enemy Secret Agents. You have to locate the enemy agent based on the provided photography, and as far as things go, this is the easiest part of the task, because the virtual streets of the Seattle aren't flooded with thousands of civilians and the search area is narrowed to a small portion of the city. The uncovered agent will start running away from you, and here's where things take a nasty turn. You can't let the escapee get too far away from you and you have to continuously fire at him, because the pursuit ends after the agent is neutralized. Because of that, those activities should be taken on only after you've invested some points into Delsin's firepower and/or have upgraded his mobility, allowing him to move around the city smoother.
The next item on the list is the search for the Hidden Cameras. After you've started the activity, the game will allow you to use the character's phone to get a glimpse of the camera's point of view - it's a very useful tip, as you can determine its location based on objects in its surroundings (if you have problems with locating the camera, stand with Delsin in some visible place and search for him via the camera view).
In the inFamous: Second Son, you will also stumble upon the most classic game collectibles - Audio Logs. After arriving near the destination of the hiding place, you must start your search. Fortunately for you, they are almost always easy to find, because the search area is narrowed - if that's not enough, Delsin can use his phone to measure the current distance from the Audio Log (as can be seen on the above screenshot).
The last group of activities are the places where you can paint graffiti (Stencil Art). After reaching the destination, the game forces a choice on you, whether you want to print a graffiti promoting your good or bad karma, so you should decide about it beforehand, based on your "path". The process of printing is trivial, because all you have to do is to move your controller in a particular motion, to paint over succeeding areas.