If you have a cube which is right next to an opponent's cube, try to stay ahead by increasing before your opponent does.
If such a cube is weaker or equal to the opponent's, try to avoid increasing: you cannot win that arms race.
Look for opportunities to increase and capture, but beware of your opponent being able to reply by starting a cascade.
Try to get control of the corner cubes first, then the edge cubes. You need fewer moves to make them reach a maximum and expand.
Try not to play too close to your opponent, especially during the opening moves. Drop back a cube or two or pick a cube on the diagonal from your opponent.
Keep an eye out for long chains of cubes that are almost at their maximum. If they are yours, you must guard them against cascading moves. If they are your opponent's, they can become ripe for capture by a cascading move, just as soon as they are close enough to your territory.
Watch some demonstrations of one computer player against another, selecting a small board and slow animation, and you will see how the rules apply and how cascading moves work (especially near the end of the game).