Normally I would tell some one to play the DLC's in the order of their release: Dead Money, Honest Hearts, Old World Blues and finally Lonesome Road.
But you say this will be the first time you have ever played any of Fallout New Vegas's DLC's. Because of this I would recomend that you play Honest Hearts first. It is the easest of the four and a good way to earn xp and gain levels. The story may be a little short but there are some amazing characters, and one of the best stories told through your pip boy to be found (explore caves that the natives consider taboo to learn what I mean)
After that I would then venture into Dead Money, but be careful with this one, there are those who love it; others who can't stand it (make a save before entering the door to this DLC just incase you become one of the later). It has amazing characters and a beautiful story all around. My advise go slowly, don't get in a hurry and have a character who is talented in unarmed, melee, guns or energy weapons. This is because this is the only DLC that will take everything that you are carrying away from you during your visit. You are given an Energy Weapon by someone and have your fists when you start. You can find knives, spears and other melee items as well as a very nice gun or two if your lockpick is high enough.
You third trip into DLC land the first time should then take you to the very intersting world to be found within Old World Blues. The story is intersting.... some like it, other's don't. The only advise I can give you is to make sure that your prefered fighting skill is maxed, you carry all your favorite and high damage wilding weapons in that class, and carry enough ammo to make the US Millitary go
. If you like hardcoe Mode, honestly for this one, and perhaps Dead Money as well I would go ahead and turn it off, just so you can follow my advise.
Finally travel the Lonesome Road, and explore the Divide in search of answers from one of my personal favorite DLC characters, and dare I say it one of my favorite FNV characters period. Arm yourself to the teeth with all your favorite toys (weapons), take enough to get by, but remeber this, this is the only one of the four DLC's that you can come and go from as you wish and if you are loaded with caps their are places to buy more goodies (new toys) and ammo. A high sneak will also help, and take your time don't rush, explore everything and everywhere the Divide holds many secrets, besides those of what this place means to your Courier.
Well there you go, follow my advise or not, this is what I would suggest for your first trip into FNV's DLC's