When photographers talk about the rule of thirds, they say sometimes they will break the rule and still the photo will have great composition. When should I stick with the rule of thirds and when should I break it? And when I break it, what should I use instead to drive eyes to specific points in the photo?
The rule of thirds to me is a rule of thumb, a reminder not to mindless frame my subject dead centre of the frame, or else I will probably end up with static or boring images overall.
As a beginner, it's a good rule to keep in mind. Not to blindly follow, but to help encourage you to try different framing, perspectives and so forth. As an experienced photographer, you'd probably not even think about it but you'd naturally tend to frame subjects off centre to make them more interesting.
Specific situations where rule of thirds might be "broken"? I would say primarily this is where symmetry is the focus of the image:
The best way to know how, and when, to break the rule is to consume photography - eat up other people's works and develop your taste, find what you like and who you like. Soon, you'll develop your eye to the point where you know how to compose a scene to best do that subject the way you wanted it.
The other suggestions are good, for giving you a general perspective of what effect certain compositions might have, but they are just the outward effects of innately being able to compose a scene from having consumed enough photography (and other art, too) of others.
For specific effects
The bottom line is that you are the boss and if you think it looks good then don't worry what the rest of us think!
Given that, here are some suggestions...
Placing a subject in the centre can:
... Or you might go the other way and place a subject right at the edge of the frame. An example is someone peeking into the frame. This makes the viewer think about what is outside the frame, and creates a not-entirely-comfortable feeling that the image is out of balance. Usually photographers want to avoid this, but if it is what you are looking for in your work then go for it.
How to drive the eye?
The human eye:
... that's a bit of a random selection, but I hope it helps.