Yes, you are probably not "required" to file. However, as a matter of convenience, you may choose to file simply because the states do not know what your deductions are and, therefore, may be expecting a return from you. Filing the returns and showing the gross income and deductions should prevent them form contacting you and requesting a return.
The expenses you mentioned - entry fees, tavel (including mileage, hotels, etc), supplies, golf balls - are all legitimate expenses for a professional golfer. If this is how you make your living (or attempting to make your living), you should report both the income and expenses on Schedule C attached to your Form 1040. If this is a "hobby", you should report the income as miscellaneous income on your 1040 and report the expenses (but only to the extent of the income) on Schedule A.