The holding period to determine whether capital gain is long-term or short-term starts on the date after the option is exercised and the stock is held, not on the date the option is granted or vests.
To receive the lowest long-term capital gains rate, you must hold the stock more than 12 months. From the starting date, reaching the same date in each following month completes another month in the calculation. The date on which the stock is sold is part of the holding period.
Example: You exercise an option on April 1. The holding period begins on April 2. You complete the one-year holding period on the next April 2.
You report capital gains (and losses) on Form 8949 and Schedule D of your IRS Form 1040 tax return, as explained in the relevant sections of the Tax Center.