Hello and thank you for choosing EmpowHer,
To put it in the simplest of terms, hyperthyroidism is caused by a malfunction in the immune system and hormonal production. Both are interrelated. As you know, the immune system is greatly effected by your nutrition and amount of exercise. Fortunately, the rules of nutrition and exercise are pretty similar for everyone.
You need to elevate your heart rate (cardio) for 20 consecutive minutes at least 3 times a week. This is very minimal cardio exercise, and more is better. Strength exercise is also very beneficial, this includes static exercises (non-cardio) such as weights, squats, ad work etc.
Exercise in general allows your body to expel toxins (sweating!!), improve circulation, strengthen the heart, expand the lungs, regulate hormones and so much more!
Secondly, good nutrition means more than simply avoiding sugar and fast-food. You did not mention the nature of your present diet, so if I am being too general feel free to message back with more questions.
But as far as diet goes, eat natural. Most of your grocery shopping should be done in the produce section. Avoid the isles of colorful boxes and the frozen food section. These are all highly processed foods which contain many toxins which you need to strictly avoid! Try to eat as many organic fruits and vegetables as possible for the foundation of your diet. Then add whole grains (brown rice, oats, quinoa etc), and lastly, hormone-free meats.
Once again, I do not know how much you know about nutrition exercise, so if you want more details, feel free to let me know.