I have faced Legendary Dragons more than any other type by a long shot. The first thing(besides their 4k+ health) is their Drain Vitality shout(most often used attack). This will remove 7 health, magicka and stamina from your character every second for 20 seconds regardless of difficulty level or magic resistance(not sure about spell absorb). There are a few ways to avoid it.
1: Given some range and the relatively slow air speed of this shout, it is entirely possible to dodge it completely. At closer range, you may not find this an option. 2:Some kind of ward. It can be the low level lesser ward you get free from Tolfdir at the start of the College quest line, something better, or the effect on Spellbreaker. Any of those will totally negate the effects of this shout. 3:Become Etherial(1 word). If you have it, you can use it.
If you do get hit, the severity of damage depends on your build. I think for most characters at this high a level a single hit won't be too bad, but a second one in a short time span could be potentially disastrous. If you have Equilibrium(Labrynthian, College QL), it may prove useful. Note you cannot cast any spells with negative magicka, even if you have gear that makes spells otherwise free to cast.
If you have Dragonrend, use it liberally(cool down is shorter than shout duration). You next steps will vary depending on your character build.
Since all dragons have 150 points of magicka they use to power shouts, a mage would do well to summon a storm atronach(or 2) and charge up Lightning storm or dual cast Thunderbolt to start. Stormies over flamers due to higher hit chance and more damage per hit, frosties will get crushed in no time trying to melee. You can switch to DC icy spear or Incinerate(ingite can work well too) when you establish the breath type and go for the opposite(PC users can cheat a little to know this on approach by selecting the dragon in the console and typing getav fireresist or frostresist, but shock is still better when they are flying around). If you're a mage who uses alchemy and enchanting, popping a custom potion of fortify destruction can shorten the battle considerably( even faster if you use the related Shalidor's Insight scroll in an emergency like an attack on a city or town). Just whale on it when you can whether it's flying around, hovering in place, or forced to ground. Shouldn't take long.
A primarily melee warrior type will get to work once the dragon is on the ground. When you have Dragonrend, they tend to stay airborne unless you force the issue. If not, it's a toss up whether they tease you from the air for a while or land by choice right away. Ranged weapons or spells are not a bad backup option. Anyway, once you get it on the ground you can approach however you like. I suggest alternating bashes and power attacks as 2H or sword and board(so you can maybe block most damage from bite attacks), or just dual wield power attacks for those so inclined(you are at risk for much more damage when a melee attack hits you even while armor capped, but you can lay on some pain pretty fast). Again, this shouldn't take too long. Your odds are pretty good.
Thirdly we have the primarily bow and/or dagger users. Atronach backups are nice if you have the option. Otherwise, wail on them with a bow from a distance when they hover or land(either by choice or by force). If you are into alchemy(and perhaps enchanting) you can pop a Fortify Marksman potion to speed things along. Probably not necessary, but can be quite beneficial when under attack in a city or town. If you are all about daggers, you might just wait until it's on the ground and run up to do a super mega ultra death combo(crouch into shadow warrior and combine a well smithed dagger with assassins blade, shrouded gloves, a power attack and savage strike in 1H). I don't know if dragon's fall for shadow warrior after they detect you(seems draugr and vampires see through it), but you might land a one hit kill even on legendary difficulty(depending on just how much of a smithing buff you have over base damage). Between a bow and a dagger, you can either take a while(safe) or get it over fast(pretty risky).
If you play a hybrid type character, you may be able to mix and match these strategies to your leisure, depending on your skill set and setting or situation. If you find a Legendary(or any other type) at a dragon lair, you may be able to get in a sneak attack to start for more damage(or get a one hit kill with a dagger before the fight begins). There are a number of spots where you can take off a decent chunk of health with a sneaky bow attack(maybe a one shot kill on lower difficulty with really good gear), but only one spot in Skyrim I know of where you can one shot with a dagger(Dragontooth Crater in the Reach. Lots of good sneak gear or shadowcloak of nocturnal helps). There are 2 dragon nests on Solstheim where a dagger can one shot the serpentine variety. They have less health(about 3500) but stronger elemental attacks(like an ancient dragon). Revered dragons are like that but weaker DV shout(5/sec) and worse elementals. A Legendary's DV shout is stronger than their elementals.
Finally, it's entirely doable to take out a legendary dragon on horseback(with a bow). I've done this a total of 2 times so far(both on this, my first legendary run) and it works quite well. Unless they sneak up on you to get in a DV shout, they are pretty easy to avoid otherwise and to circle strafe while hovering or on the ground. You don't even have to get off the horse to absorb its soul. That's pretty cool. You should dismount for looting, though. Happy Hunting.
P.S. If you dual wield, it is possible to "one shot" a legendary on lower difficulties(given well smithed and/or enchanted swords and some good fortify onehanded gear) once your prey is grounded. For 1H/Block or 2H, you're facing at least a minor slug fest. Hope this proves helpful to some folks.
Edit: Regarding avoidance and keeping horses/followers alive. First, avoidance.
If you are traveling about, try to keep your distance from any known outdoor word walls. Sometimes dragons just show up. Sometimes they engage pretty much right away, so there isn't much you can do. Sometimes, they just fly around a bit and then fly off somewhere(often to a word wall, but not always).
As for keeping horses/followers alive? Let's start with horses.
If you intend to enter combat with a dragon in the near future(approaching an outdoor word wall location), dismount a fair distance away. The battle should take place closer to the word wall and your mount should be out of range. If you get ambushed on the road or in a field or something, keep on the move. Hopefully you have a decent bow. If your character doesn't use real bows(not to say conjured bows aren't "decent", but you can't use them from horseback) then I'm not sure what to tell you. On to followers.
Ideally a follower should be well equipped with enchanted armor that fits the way their class fights, and likely a well smithed weapon or two as well if the follower in question isn't a pure mage. If they are a mage they should have plenty of extra magicka and regen enchanted on what they are wearing so they can keep slinging distruction spells and/or conjuring stuff like atronachs. Your combined might should take down the dragon fairly quickly. Hope this helps.