Sure! Healers have the job of healing their group, buffing their group, and if needed, doing some DPS. Any class can be a good healer.

So the basic tool of a healer of any class is the restoration staff. For basic healing, Grand Healing/Healing Springs is a great go-to AOE heal. You can supplement that with other restoration staff skills or class skills, but that skill will be your bread and butter.

The typical healers you'll see in group content are Templars and Wardens. Both classes have a lot of dedicated healing skills. If you want to run a stamina healer, definitely go warden. Most other healers will be magicka builds. However, any class can be a healer. I had a rather effective MagDK healer in PVP using healing springs and mutagen. Templars and Wardens just have a lot of specific healing skills as class skills more so than the others. A mag sorc healer can be extremely effective with the Twilight Matriarch heals and a resto staff.

Once you've got your heals figured out, you can look at buffing your group. Again, all classes have options for buffing their group. Magicka healers will typically run at least one bar as a destruction staff for Elemental Drain with its magicka steal. I prefer a lightning staff as it also can produce the off-balance effect for my DPS. Its also nice to have a DPS option, because there will be groups that you can keep healed just fine but the DPS is so low that you are desperate to help them out a little.

Tips for gameplay:
1. start easy on yourself, healing some of the easy normal dungeons and working up to the harder ones. You'll get a feel for hard-hitting mechanics and how to deal with PUGs.
2. Proactive healing is better than bursting after-the-fact healing. I tend to keep a heal-over-time skill like Healing Springs or a Warden/Templar skill active on most of the group as much as possible. That way I don't have to waste magicka spamming big heals when somebody drops to low health suddenly on a big boss health.
3. You'll learn very quickly how competent your group members are and adjust accordingly. Before long, you'll recognize very quickly who's way too squishy for the stunts they want to pull and who's a fake tank and who doesn't need your heals at all and who is perfectly capable of burning the boss but does need your heals to stay alive. A healer almost knows more about how competent the group is and how well they are built for their roles than anyone else.
4. Be prepared to deal with fake tanks in the group finder and thus be prepared for sometimes having the boss in your face because someone couldn't be bothered to hold aggro. I main a tank, so on my healer I'm perfectly comfortable holding still and healing myself through the boss's attacks so the DPS can burn the boss. However, if you feel like you need to run like a chicken with your head cut off to survive, go for it and yell at the fake tank afterward.

Gear:
Trials level gear: Spell Power Cure, Worm's Raiment, Sanctuary
Dungeon level gear: Julianos (crafted), Seducers (crafted), Eyes of Mara (crafted), pretty much any light armor dropped set. Honestly, for dungeons, especially normal dungeons, you should be fine to slap on most any light armor sets and be fine.

Tips, you can pick any magical race, but Argonian is by far the best if you want to dive in to PvP and back to PvE, maybe turn it into a tank. (I made a breton and yeah it sustains fine, but I can't take him into PvP because he can't sustain stam and dies easily compared to Argonian.)
Then I completely agree with above. 1 Start on easy/normal dungeons.
2 Proactive heals are crucial, wasting Breath of Life, killing your magika isn't smart. I have 4 heals over time, even if anyone drops down 8k health, it's back up in almost a second. Breath is only an "Oh Crap" move.
3 Yes, as above, be mindful of your group and don't be afraid to talk to your group by saying "get out of red" "stay close, I can't heal you back there". Then be mindful if you see your team struggling, throw them shards/orbs to help their resources.
4 yes, there will be fake tanks, even bad dps, but just do your part as best as you can. It already sucks having a bad tank, don't be a bad healer, keep calm, heal through.
5 Trust in yourself, mistakes are okay and normal dungeons is where we learn and can make mistakes.

The sets suggested above are great, but for new player friendly or low level.
Sanctuary will be the first one you can get at level 10 and is super strong, pair it with gossamer and your group will be hard to kill, everything is easy mode.
But worm and spell power cure will be your go to for competent groups.

If you are running a Sorc healer (and it can be done well) then be sure to work up your pets; the Twilight Matriarch is a great emergency heal and it's not much limited by distance or direction (the Templar Breath of Life was pretty directional for a while; doesn't seem to be so much lately tho).

You might also give thought to putting the Healing Staff Ulti on at least one of your bars.

Be sure to take a look at the Undaunted line; Necrotic Orb is one way to allow allies to get some Stamina or Magicka back.

I know some folks get good mileage out of Blood Altar and it's morphs, but I've found the synergies seem to be really hard to hit.

As others have said you'll generally see Templars and wardens as healers but any class can heal (with various pros and cons). Personally I was a Templar until wardens came along, and I haven't looked back since. Once you find a class that you think you'd want to heal on, then it becomes easier to narrow down builds. For me I think about tempers and wardens this way:
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)

The #1 thing I could say overall though is anticipate damage, don't respond to it. Keep healing springs out, keep mutagen up, and watch for what the boss is doing. Just as much as the tank, you need to anticipate when an attack is coming that you'll need to respond to. Mitigation works the same way. You don't have to heal damage you don't take, so keep those protective buffs up well. If you do miss something (and we all do so don't sweat it too much) be ready with a burst heal. Whether it's breath of life, combat prayer, budding seeds, it doesn't matter which. Just know whether you'll have to aim it or not (which brings me to tip #2)

#2 is positioning positioning positioning. Placing yourself strategically is critical for successfully reaching folks with heals, and buffs. Sometimes ranged builds want to stand back 500 ft so they're safe. That's fine. They're also impossible to reach with heals or (arguably more important) buffs! Don't be afraid to let them know to move, or reposition yourself to see the entire battlefield and group.