Tanking is a little bit weird in ESO. While there are several important aspects to it - keeping agro on boss and hard hiting ads, cc-ing trash(DKs are particularly good for that because they have Chains to pull ranged ads in and Talons to keep them all in place), keeping everything in one place, debuffing enemies/buffing your team as much as possible(there're sets adding to their health/reducing monster's defence/etc, s&b taunt provides a very good defence debuff, Warhorn ultimate buffs team's resources and damage, etc), it's entirely possible to do all normal and most vet dungeons without a tank completely. With a tank, it is kinda boring in easy instances - you just taunt and stand there for the most part, watching enemies melt. Tanking is probably my favourite role in harder dungeons though, ergo vet dlc ones. That's the ones where you actually need to know mechanics and follow them carefully(especially as a tank) in order for everything to work. That's where you gotta watch for cues, move, re-taunt, reflect, dodge, interrupt, turn away etc etc just in time or it's a wipe. I enjoy that.

Tanking is also probably the role most dependant on the group, as weird as it may sound. As a good tank, you should pretty much never die short of mechanical one-shots, even without heals, however as a good tank you're unable to add(much) to dps and all the buffs go to waste if there isn't much to buff in the first place. In a great group, things can melt so fast a tank is barely needed. In a subpar group, an easy run can become lots more fun due to fights taking longer, people being more squishy, everyone but you dying, etc, and it can make for a fun challenge, but at the same time a run can fail completely due to not having enough dps - and there'll be nothing you as a tank can do about it.

Far as classes go - templar is the go-to for heals due to best burst heal in the game(Breath of Life), best resource management skill in the game(Shards; now there's also the Orb from Undaunted skill tree, but shards restore both resources and they have a much more reliable synergy), cheapest purify in the game, etc. However, any class can heal and there's a slightly different "feel" for different classes. Sorc, for example, also have a great burst heal, but it's dependant on their pet being alive, and if pet dies you need like 2 seconds to summon her again, which can be unacceptable in some fights. DKs...DKs used to have great group shields but they've been nerfed so many times I'm not sure they're any good anymore. I haven't run with a DK healer for a while. NBs have great off heals, they literally off heal quite a bit while just dpsing. Wardens have great HoTs and some nice group buffs. Honestly if you aren't going for the absolute FotM you should just try for yourself and see.

The go-to tank is DK thanks to the amazing CC and amazing passives. There's currently no stamina regeneration while blocking and DK has hands down best ways to recover stamina while blocking. NBs aren't too bad also I think - NB is one class I don't have though. Sorc have a nice CC for keeping trash in place and a nice skill letting them convert either magicka into stam&health or stam into magicka&health, but that does require you to drop block. Templars are kind of screwed in terms of stamina restore, but I love tanking on my templar. Mine is a health build relying on a cool skill templars have - Blazing Shield, a shield that scales off your health and damages enemies when exploding. I can offheal on her really well due to high magicka recovery and BoL but I'll admit managing stamina is about 25 times harder than on a DK(which doesn't mean it can't be done, have no death tanked vet dlc dungeons just fine).

Tl;dr - best tank is indeed DK, best healer is indeed templar, but anything can work if you're willing to put in the effort You'll never know til you try.

Healing is very interesting in ESO.

Before even discussing classes/rotations, it is VERY important to note that there are NO targeted heals (ones cast at a SPECIFIC player chosen by the healer) like in many other MMOs that people are familiar with. It is all AOEs and "smart heals" (that default to the lowest health character). What this has come to mean is that healers get their AOE heals down, and their heal over time buffs going on the group, and then use burst heals to support whatever player drops the lowest. Sometimes these burst heals are smart heals (like the breath of life skill a templar has, or healing ward from the restoration skill line) or some are directionally aimed heals (like fungal bloom for a warden, or combat prayer from the restoration skill line). In this way healing is as much about keeping the buffs/AOEs up, as it is anticipating and responding to burst damage the group takes.


Looking at the classes then:
As others have said, any class CAN heal, but most often you will see Templar or Warden healers. Part of the reason why any class can heal, is that many of the skills a healer uses in fact come from the restoration staff skill line, with their class skills augmenting and supporting these abilities. Templars and wardens tend to stand out though both for the group buffs they can provide, and their raw healing ability via their class skill lines which directly support a healing role. Templars also have the unique ability to restore both stamina AND magika to a teammate via their Shards ability, while every other class relies at least in part on the Obs ability (from the undaunted skill line) which is harder to aim, and only restores one of the 2 resources.

Having played both warden and templar healers (I cant speak to Nightblades, sorcs, or DKs for healing), I tend to think about them in 2 ways:

Templars build a healing house (setting down AOE heals and buffs around them) and support/protect it. If they move too far, they need to rebuild the house when they get to a new spot. Healing wise they have very good smart heals, but their targets are limited and some heals are expensive (or have a cast time). Their group buffs though are unparalleled.

Wardens on the other hand are mobile and throw their healing. The big buffs a warden has move with them (such as the netch or the protection from ice cloak) or are directly cast on others (leeching vines, fungal bloom) but the price they pay is that their buffs and heals must be aimed. This means moving and looking constantly, but on the plus side our big heal (budding seeds) can hit as many targets as we can get in it's area (again, proper aiming required). The resource sustain a warden has, and raw healing they can put out is almost unparalleled though.
Otherwise both classes share heavily in their use of restoration staff, destruction staff, and other support abilities (Purge, war horn etc)



Whichever class you ultimately decide to play, youll still be essentially doing the same things, just with a different flavor and playstyle. Welcome to ESO and the world of healing! If you have any other questions about particular builds/abilities just shout!



Edit:
For full disclosure I currently spend the vast majority of my time on my warden healer. The templar is still a great (and many people often will say best) healer, but in terms of raw healing power, and interesting gameplay, to me the warden takes the cake. Yes the healing can be a little trickier (no breath of life instant smart heal for someone low on health) but I dont mind having to aim my heals or anticipate damage, and I love the mobility offered by being able to throw my healing circles away from me, rather than having to re position myself each time. I find it has made me a better healer even when I do swap back onto my templar.

That said, for someone getting used to smart healing and laying down AOE/HOT spells and then supporting it with burst heals, it is VERY hard to beat a templar in terms of ease of use, especially at the lower levels before a warden has their full arsenal of skills. For this reason I personally do recommend starting a templar healer, and then once you have the hang of healing on them, look to spread to other classes where it may require more situational knowledge and experience to be as effective. Just my 2 cents.