The air-driven artificial horizon is subject to acceleration and turning errors, which occur because of the precession of the acceleration forces applied to the pendulous vanes and their housing.
In a typical GA training aeroplane, the acceleration is insufficient to cause much of an error; on a fast-jet type, the error might be significant and indeed led to a few crashes on Meteors, Vampires, etc.
So, for all intents and purposes, in a GA training aircraft, the AH can be assumed to be error-free, apart from the self-induced possible error (mentioned by RatherBeFlying) of mis-setting the miniature airplane symbol.
Of course, the AH may topple if an extreme attitude is achieved, and in this case you are better off referring to the turn indicator and pressure instruments.
I would recommend the T-scan and other scans alluded to in previous posts. Although the AH can be regarded as the “master instrument”, the other ones play an important part in confirming whether or not a particular manoeuvre or performance target is being achieved.
I also endorse heartily the “Attitude plus power equals performance” mantra quoted by previous posters. I would add, that for more advanced I/F, you could add “configuration” to the left of the = sign.
Although I agree that the PPL syllabus should concentrate on visual flying, an understanding of the instruments and a working knowledge of some basic scan patterns will do no harm.