Do not be afraid to use a whip if needed. It is only used to emphasize your leg aid, not as a punishment. Use it behind the leg only after asking with your leg first, and the horse will become sharper to the aids.
I like what I hear you say about your instructor, it is important to right away begin to understand contact, for you & the horse. I like to explain contact as a living communication with your horse, without contact, the aids are unclear. It is also good that she will get on your horse to show you something or to check on the horse's response. Very good!
If your horse was a western horse, if may be used to going slow and without suspension. Wake him up & get him moving. It is better to see a horse going a bit too fast than puttering along. Quarter horses tend to do well warming up with lots of canter work instead of trot. It helps to soften & loosen their back.
Go to shows and watch the upper level riders to see what you should aim for. Always tape your lesson if someone is available to do so, you can rewatch it before your next ride to help you learn. Often times I understand what the instructor is saying by watching myself ride while listening to the instructors comments & instructions.
Dressage is a great sport because you are only tested on what you have learned & acomplished, not in comparison to anyone else.