Is the exercise of my stock rights taxable to me ?

You do not realize a stock gain on your exercise of stock rights. A capital gain or loss will be calculated in the year that you sell the stock.
The holding period of the new stock is as of the day you exercised the rights. The holding period will be used for the short term and long term treatment of capital gains and losses.
Your cost basis for the new stock will be the subscription price you paid plus your basis of the rights exercised.