This is easily the most complicated subject in racing. It is thus very difficult to comprehend from the driver's seat. Weight transfers are something you learn for a particular car. They are the thing that you dial in on the chassis during practice. Point is, you'd better know what's going on and what's going to happen before race day.
Weight transfers are complicated because they are determined by a hundred changing factors, and they are driven by pure physics, something not easily contemplated when accelerating at 1g.
Thus, I'm not really going to try to tell you all the different weight transfers that can possibly happen, as there are far too many, and far too many outcomes depending on the car and your driving. What I will tell you is this, your car will rely more and less on different wheels in different situations, and you'd better have a good idea of what to expect.
Weight transfers are really misnomers, but lots of racing is misnomers, so it's ok. Weight in this case refers to force, as in the force that a particular wheel is exerting down on the ground, which changes its grip. The other two dimensions are expressed by the traction circle. Too much on a wheel, it will lose grip and spin. Too little and it will exceed its grip and slide. Both are bad, so you try to keep them at a minimum.
You minimize sliding by trying to balance the weight, which means getting it to the proper part of the car, where it can be most used. Again, the proper weighting for each car in each part of a turn is very complicated. A front-wheel-drive car is a compulsive understeerer, a light midengined race car will trade understeer and oversteer very quickly depending on the driver input.
There are Four Rules of Weight Transfer, Three lesser, one greater:
Lesser the First: Turning the car will weight the outside wheels heavily, the inside wheels lightly.
Lesser the Second: Accelerating the car will weight the rear wheels heavily, the front wheels lightly.
Lesser the Third: Braking the car will weight the front wheels heavily, the rear wheels lightly.
Using these three lesser rules in combination, one can balance the car, or at least have a clue as to the intentions the car has for you. Your driving techniques (such as heel and toe and trail braking) are developed to take advantage of weight transfers so following them will properly propel you through the turns. You must follow the traction circle, of course, it is your guide. It will tell you when you have exceeded your traction in a very impolite way. But all the lesser rules are common sense. The one Greater rule is the key to driving success, it will never fail you.
Greater Rule the Only: The car is changing, you must change with it; be smooth in all motions, in all driving.
This sound stupid? It is the key, damn you! A road course is like a giant physics equation you will work out in real time, one too complicated to comprehend by anyone but the almighty programmers of Gran Turismo! But you needn't comprehend it to solve it. You must follow the car, feeling it shift around, and sense its limits. The only way to do that is to be smooth. Sudden motions will transfer weight from one corner to the other so quickly you will not know what the car wants to do, and you will not be able to react quickly. Smooth motions allow the car to tap you on the shoulder politely and say "I'm about to crash into that wall unless you slow down," or "I think you have confused the accelerator with a pellet dispenser and I will now spin."
The most important thing about driving is understanding that you are in control (or should be) of a machine that was made to accept certain limits and tolerances, and you are purposely trying to go deep into those limits without exceeding them. Balancing the car with smoothness is your best bet to remain in control of the car, remain ahead of the pack, and remain safe.