Ok, a very quick summary is :
Consider Trials like a veteran dungeon but 3 times bigger, more waves of mobs and tough bosses in stages. Usually gear will be similar to those used for vet Dungeons, but a lot of people gear up for group support especially healers and tanks. Eg they have gear like Robe of the Worm which reduce the magicka cost of skills by 8% etc. Often these drop in trials so you might need to run them a few times to get some. The normal version eg Normal Maw of Lorkhaj is much easier than the veteran version.
Check the Undaunted line for group skills and synergies. The alliance lines can help for a healer eg Aggresive Warhorn that buffs your party can be very good for trials

We've been running trials in our guild for a few months, these are my notes on what you need to know/have beforehand:

1) Join a guild that does trials: Trials require a hell of a lot of coordination, and it's impossible to do them with people you pick up in the zone. So join a trials guild, or try organizing trials in one of your own guilds. Usually it's a bit difficult finding groups (even in large guilds) in the beginning, since most people have never done them either, but more people start joining once the trials become popular.

2) Voice chat is a must: You can't coordinate 12 people without voice chat. Ignore all those idiots who want to join but say they will not join voice chat under any circumstance, they'll be the first ones dead. Not everyone necessarily needs a microphone, but definitely need to listen in.

3) A raid leader: Someone who is familiar with the mechanics of mobs and bosses in the trials. A good raid leader should be able to explain all the mechanics of the trials before each fight. A good raid leader will also perform pre- or mid-trial checks, like if everyone is geared properly, whether everyone knows their roles (how to stand, when to cast skills, etc.), has appropriate skills (buffs or debuffs) for certain mobs/phases, and that no one is going in with broken armour and no buff foods. Raid leaders are often difficult to find, but you can become one yourself if don't find anyone. Just read up a lot on trial strategies online and then go in with your group, often it's a lot more fun learning on the job!

Other than that, the first trial you should go into is Aetherian Archive, this is by far the easiest of all four trials available. After you finish that, go to Hel Ra, then Sanctum Ophidia, then MoL ...

Healers are usually required to slot lots of support abilities, like novas, spear shards for tanks, elemental drain, etc. Again, it's much easier if you just go in and have a look around, you'll figure out what you need.

Also, if you're on PC/EU and would like to join a trials guild, let me know. Always happy to take more healers along with us