Is there an order to playing Skyrim and the DLCs?

Both Dawnguard and Dragonborn have new map areas added. When you start dealing with named vampire NPCs, that's Dawnguard (although there are a couple named vampires in the main game, but no major plots). If you traveled to Solstheim, that's Dragonborn. The DLC quests are independent of the main quests so you can't accidentally mix them up.

Hearthfire don't really have quests. It has extra custom houses which are significantly better than the vanilla ones.

As for using them, you really should have all your DLCs activated at the start, before you create your character, since converting your save file at a later date can cause unforeseen issues. However, if you have already played for a bit with the vanilla game, it is still playable to convert later, since these official DLCs expect you to have an existing character.

I also recommend looking for the unofficial patches as they resolve multiple issues not fixed by the official patches.

In general, each DLC adds content to the game and they can all be safely installed and played at the same time. The fact that some of the foes you'll face in the wilds of Solstheim are a little tougher than those on the mainland may encourage you to face the challenges of skyrim-dragonborn a little later.

I don't generally focus specifically on the main story lines in any order, but I frequently find myself finishing the dawnguard story-line before I finish the main Skyrim story line ...

...to both cut down on the muggings by the faction I'm working against and because I find that uncontrolled vampires do more damage to the populations of my cities than do uncontrolled dragons.

The Hall of the Vigilant

The only Change that may leave you feeling that you've missed out on something is that loading skyrim-dawnguard and reaching level 10 will cause the Hall of the Vigilant to have been sacked by vampires. The location does not seem to have any quests to which you'd loose access. Though a player that intends to specialize in Heavy Armor may wish to visit for a single point of training from a book, The Knights of the Nine. There is also a Daedra Heart laying about which is rare enough to consider visiting before it's sacked.

Entertainingly enough, there is also Keeper Carcette, an Expert Restoration Trainer there to serve the 0-75 Restoration training needs of your pre-Level 10 characters. Not much of a loss as Danica Pure-Spring, of Whiterun, is a Master trainer (0-90), as is Florentius Baenius, who is added by the same DLC that kills off Carcette.

Another interesting skyrim-dawnguard change is that...

...folks don't get violently aggressive toward you if you allow yourself to become a stage 4 vampire. Personally I think it's a shame that you no longer need to manage your "hunger" if you're careless enough to allow yourself to be turned by a common vampire (not to be confused with the much smarter/cooler Vampire Lords, who will keep a stable of "Cattle" for you to feed upon).

Activating any DLC will not affect the original storyline.

Environmental changes:

You may get attacked by the followers of the dragonborn and get different responses for being a vampire.

If yes, in what order should I use the DLCs?
I recommend doing the main or side quests first. You'll get the required items and levels needed to conquer the "harder", required parts of the DLCs.

For the later parts of Dawnguard you need to have

the Elder Scroll

from the Main Quest.

And I highly suggest not going into the Dragonborn DLC until later in the game (Traveling to Solstheim and further adventuring) because I had problems beating later enemies which were essential to the Quest. So far so good, I will come back later and have my revenge! Time to farm some dragons!

Seemed like a fun idea... Until the antagonist came and stole each Dragon Soul. This may occur after you met him the first time.

For more information on WHEN to start the DLCs have a look at this question. If you feel ready to start with the DLCs, the order of Dawnguard or Dragonborn does not matter. Feel free to do Hearthfire anytime you like, as it doesnt require any power.

If no, can I use them all at once?
Yes, as with most other Quests you can play all of them at the same time unless:

You are in a different dimension. In that case you have to finish the mission that got you there first.