Listen, pal- horses don't just start acting up like this for no reason. There is ALWAYS a reason why, and it's up to YOU to do some detective work and figure out what that reason is. It sounds to me as though this mare is either in pain ( whether in her mouth, her back, or her feet) or she has been abused in some way and this is how she is responding to that abuse. NO horse can rear and buck at the same time- these two activities require different muscles, and shifts of balance. Rearing is most likely to happen when the horse is standing still, whereas bucking is something which happens while the horse is moving. I have never seen a horse which could buck at a standstill without losing his or her balance- so I take leave to doubt that this mare is doing both of these things at one time. I think you are exaggerating on that point- either that, or you are a TROLL who is asking this to get attention. I will proceed on the assumption that the former is true instead of the latter.
Clearly, the first thing to do is get this mare thoroughly evaluated by a vet to rule out medical problems, and while you're at it, have her looked at by an equine dentist and chiropractor as well to rule out mouth and back pain. Next, have a master saddler come and check the fit of your tack ( look in the phone book and ask around for the name of a saddler in your area) to make sure that your saddle is not pinching the mare's withers or putting pressure on other parts of her spine. I wouldn't rely on your trainer or instructor to look at the saddle fit, because most trainers only rarely if ever have the knowledge to do this correctly. It may turn out that the saddle needs to be reflocked or restuffed, and a saddler can help you with this. Lastly, you need to have the mare looked at by your farrier to make sure she isn't having issues with her feet and shoes which are contributing to her behavior.
If everything checks out medically, and there are no other issues which are causing problems, the next thing to look at is the mare's environment and routine. Does she get any turnout time when she's not being ridden? What is she being fed, and how much? Is she on a diet that includes a lot of sweet feed or alfalfa hay? If the answer to that last question is yes, then you need to be aware of a few things.
There are some breeds of horses, particularly Arabians and those of Arab ancestry, which cannot be fed large amounts of sweet feed. Most commercial sweet feeds for horses contain large amounts of molasses,which in turn is a by product of the sugar refining industry and is very sweet. Arabian horses, and other breeds with Arab blood or genes in them often have trouble metabolizing the sugar, and it tends to make them "hot" and hard to ride and train. This can be compared in some ways to what happens when little kids eat too much candy or drink too much soda- they get a sugar jag and can be impossible to deal with. From your description of your mare's behavior, I have to wonder if this might be an issue with her which you need to look into and correct if necessary. There are lots of lower energy, nutritious feeds on the market which do not contain such large amounts of sugar- we feed our horses beet pulp for this reason, something you may want to try yourself. Next, about the alfalfa. What kind of hay does your mare get? If it is alfalfa, be aware that a similar situation exists with this kind of hay, just as it does with sweet feed. Alfalfa is a very rich hay, full of protein ( it has the highest protein gradient of any hay which is fed to horses, anywhere from 14 to as much as 18 percent or even higher, depending on where it is grown and baled) and too much of it can not only make horses hot and lead to behavior problems, it can also cause the animals to founder or colic. For this reason, Timothy hay or a Timothy alfalfa mix is really best for horses to eat- and in the case of your mare, that mix may need to be Timothy and grass until she settles down.
I would try altering this mare's diet and routine, and make a point of longeing her for at least 30 minutes a day, BEFORE you get on her back. She needs to have a chance to buck, play, run around, and get the fidgets out before you get on her, and longeing her is a great way to accomplish this. Make sure she gets at least some turnout time every day too- she needs a chance to relax and be a horse, just as you need mental down time to stay healthy yourself. Turning her out will keep her from getting bored in the stall and resorting to vices such as cribbing to amuse herself, and it will mean she will not be quite as fresh and frisky when you do go to ride her.
Good luck- I hope these tips help. If all else fails, and the mare is still acting up at the end of a month or so, I would consider getting some professional help for her.