YAY! You chose a good airline for your first flight.
Yes, your backpack will be considered a personal item and fits the size constraints. It should be placed under the seat in front of you. Since it is right on the limit for size, I wouldn't overpack it as it must fit completely under the seat.
Carry-on bag limits are 24" x 16" x 10" and yours is under that limit...so you should be fine with that too. It will need to be placed in the overhead bin.
Just in case your carry-on bag needs to be gate checked due to no space (it sometimes happens), I would recommend that you place medications, jewelry, sunglasses, or any other valuables/non-replaceables in your personal item (backpack).
Enjoy your flight!
