I am 22 years old. And I have size H breasts. Bra size 34H.

It sounds like you would benefit from breast reduction surgery, and achieve relief of your symptoms.  The size you would like to be is something you should clearly convey to a board certified plastic surgeon who is very familiar with this procedure.

Thank you for the question. Based on your description, it is likely that you will do well with breast reduction surgery; this operation, is one of the most patient pleasing operations we perform.  Most patients undergoing breast reduction surgery wish to achieve enough of a reduction to help with their symptoms while remaining proportionate with the remainder of their torso. 
With the goal of improving communication with my patients (determining their goals in regards to breast size after breast reduction surgery) I find the use of photographs of “goal” pictures (and breasts that are too big or too small)  very helpful. I have found that the use of words such as “natural” or “C or D cup” or "proportionate” etc. mean different things to different people and therefore prove unhelpful. 
Also, as you know, cup size varies depending on who makes the bra; therefore, discussing desired cup  size may also be inaccurate.
Key to success: select your plastic surgeon carefully and communicate your goals carefully as well. Again, in my opinion, the use of as many visual aids as possible is helpful. I hope this helps.

Having a breast reduction can completely change your life and the size will ultimately be your choice.  I think it's important to try to stay proportionate tot he rest of your body.  According to your height and weight it sounds like a C cup would fit you well and help you feel less uncomfortable during physical activity.