My advice would be to read through some good tomes on Burgundy, like the Hugh Johnson's World Atlas of Wines chapters on Burgundy, paying special attention to the text in conjunction with the maps. That's especially valuable, because the maps give you the indications of why the villages might be different in the first place.
Then, go to Burgundy. You can either go selectively from one village/commune to the other (but for godsake don't stop at all of them, just the ones that most stimulate your imagination or sense of wonder) and taste judiciously. I'd actually recco that you stay with the village designations through the Premier Cru. Less damage to the pocketbook that way.
The other alternative is to go to Nuits and to Beaune, and taste selectively in those two towns, because each will have places to taste wines from all over those areas. Especially Beaune, which will have the widest selection of all of Burgundy. Curiously enough, there's a major tasting room (where they will also twist your arm to buy buy buy, but what else would you expect?) right next to the Visitor's Information Center and the Hospices de Beaune in the heart of the city.
If you have enough time, don't neglect the Cotes Chalonnaise, which a lot of people do. To get the full sense of Burgundy, what I've done is start in Macon and work my way up all the way to Fixin---but that took a minimum of three days, was a professional tour, and was damned exhausting, fun only to the demented or the idiot who's trying to learn about Burgundy in the shortest way possible.
A lot..most everything, as a matter of fact...becomse much clearer from the simple fact of being there (I like to watch.), and seeing/feeling/sensing the climate and topography, and seeing in full three dimensional glory the layout of the vineyards and where the PCs and GCs are located. It triggers that moment of awed comprehension when you actully feel like you understand why things are the way they are (you don't, not really, but you feel it for a moment).
For me the thunderbolt of comprehension, the single most meaningful moment in my understanding the basis of Burgundy and the entier AOC structure and hierarchy, was standing between Chassagne and Puligny, pretty much at the base of the slope, looking up the two lane road that cuts between the two AOCs (where St-Aubin is) heading up to the Haut Cotes de Beaune. I could see everything I needed to see to understand---to understand the subtle differences between Chassagne and Puligny, to understand why St-Aubin was different (good but never more than PC simply because of where it was), and to begin to understand all the little patchworks of Puligny----Le Montrachet, Batard, Pucelles, etc.
So what I'm telling you is, go and experience. And look out for the thunderbolt.