Perhaps the work without stirrups should be done in walk first though as starting of without them in trot could make the rider tense up and lose their balance.
In walk try doing a few exercises without stirrups to improve your balance

1/bring knees up and then streach legs down
2/ bring one leg back from the knee then the other then streach legs down
3/round the worlds ! a childs exercise but gives confidence. has to be done while horse is stationary to begin with though This is where the rider does a complete 360 turn bringing one leg over to one side so both legs are together ,then faces back of horse then the other side and finally back to the starting position. Hope I have explained this ok.
4/ standing up in the stirrups bottom off the saddle

These exercises can be done at first keeping one hand holding on to the front of the saddle for balance and once the rider is more balanced and confident they could try some of the exercises not holding on or even in trot.

I am a great believer in hacking out too as sometimes a lesson in a school can be hard as there is a lot of " steering "to think about. On a hack it is more relaxed and therefor the rider can concentrate on enjoying the ride and can think about feeling the movement of the horse without worrying about corners, what their legs and hands are doing etc.

Do you ever get to go out hacking ?

In trot using the hand on the saddle for balance is a good tip , pulling the seat down in to the saddle , relaxing the leg and pelvis and doing a sitting trot. A very good way of feeling the movement , pretend bottom is glued to the saddle
Posting or rising to the trot can be confusing as some riders just think that you count 1 and stand up and 2 sit down but they get the "beat" wrong and that is why they bounce. Dancing to a twist instead of a slow song or vice versa Learning to count out loud to the trot and watching the horses shoulder movement ie sit when the right shoulder comes back can help explain when to sit and when to post/ rise.Also ask the teacher to bandage the horses legs ie off fore and near hind then look at the way the horses legs move in the trot.

Rising by standing on the stirrups is not exactly what a rider should be doing as the movement of rising should actually come from the pelvis and a supple but strong back and it is more of a thrust forward than a standing up movement . Standing in the stirrups can cause a rider to come up too high lose their balance then miss the beat and again this will cause a bounce.

As you can see relaxing, suppleness and balance are very important in riding so as mentioned the exercise ball thing is very good for achieving these things. There is actually a book on riding that tells you how to use the exercise ball to improve all these things but cannot remember the name of the book I shall have a look and see if can find out.

So relax enjoy and learn