HIT THE TOWER
Let's start with the basics.
In order to win a game of League of Legends, you need to kill the nexus. Before you can kill the nexus, you must first destroy the towers and inhibitors between you and the nexus. Since there is only a single lane in ARAM, every tower that you destroy marks a significant milestone that literally brings you closer to victory. Therefore, in some sense, inflicting damage on a tower is the only way to actually make progress in ARAM.
This simple observation highlights the most important mindset shift that you can make if you want to get better at ARAM. The champions on the enemy team are not actually your enemy. They are simply obstacles between you and the true enemy - the towers. If you see an opportunity to hit the enemy tower, then you must take it. If no such opportunity exists because of the aforementioned obstacles, then you must create a plan to remove those obstacles.
With that in mind, let's define a simple heuristic for our decision-making framework. We will prefer strategies that allow us to maximize damage to the enemy tower and minimize damage to our own tower.