My "baby healer"

Just about any class can do some sort of heals, but the easiest/earliest by far are from just playing with a resto staff. In a perfect world, I would want to offer some resources too - that can be tougher.

Heals
Resto Staff
1-Grand Healing, morphed as soon as possible to Healing Springs 'cause magicka regen is where it's at for healers
2-Regen, morphed as soon as possible to Mutagen 'cause heal more peeps
3-Undaunted Blood Altar - lot's of folks think of this as a tank skill, but anyone can throw it down, especially where there isn't much maneuvering room.
4-Strife - a NB skill that quickly morphs to heal other party members
5-Consuming Darkness - a NB ulti that does some decent group healing
6-Armor sets can help this; Combat Physician (from Wayrest) gives bonus heals with crit healing; Sanctuary (from Banished Cells) boots healing received. Or ColdHarbor's Favorite, but you might need a really crafty friend or guildmate to get a set.

Resources
Rune of Protection (fighter's Guild) at least gives stamina help in all it's morphs
Necrotic Orb - takes a while to get so it's not really a "baby" skill
Lamia Armor (from Arx Corinium); gives health, magicka and stamina to nearby party members on use of any Ulti. Can boost any healing ulti, or make any ulti contribute to not only healing but resources.

I level chars as tanks and healers a lot of the time because it's easier than taking up a DD slot when levelling.

Levelling Tanks:
I use Adept Rider and Fortified Brass to level them up. Having the speed to get there first makes tanking easier because too many people forget first person to damage a mob gets initial aggro on it (keeps the groups together while you worry about taunting things).
Sword and Shield with Pierce Armor and you're halfway there. I know people go on about silver leash or unrelenting chains but honestly, if you're new to the business you do not have the resources to pull things. You wont see a high return on stamina or magicka from CP because you wont be getting the killing blow to trigger it.
The second half is physician, heal thyself. Ensure you slot some kind of self healing ability so you dont die if the expected 'fake healer' occurs. If your class lacks an option, unlock Vigor from the pvp line (by entering a battleground and doing what you can without any semblence of an idea what to do, if that's what it takes).

Pierce Armor and self healing and you're good to go.

When you hit 50, things change and you can begin to look at proper tanking, such as rounding up mobs, but for levelling... don't bother because you just don't have the resources to handle it as quickly as they die.

Levelling Healers
I use Adept Rider and Order's Wrath to level these. Having the speed to keep up with the group makes healing easier because panic is the easiest way for people to die.
Restoration staff will cover all of your needs for healing, so everything else is just to flesh out your kit later. But to start with Blessing of Protection + Regeneration is all you need. Some classes have other heals to support this kit but realistically, those 2 heals are all you have to worry about - 1 is a nice hot over people while the other is a direct aoe burst.
Don't be afraid to heavy attack a lot for resources, don't let your magicka get below 50% because you fear not contributing to dps is an issue. Eventually you can hit 50 and make hybrid healer specs that push 20-30k dps while still capable of hitting 10k hps if the fight suddenly requires it.

These days Ice Staves are the go to staff for healers on the backbar, to provide the shield when using blockade.

When you begin starting out at 50, there are plenty of existing guides to follow so I won't add much about that except the number 1 key to being a support character is not letting rubbish DD get away with acting like children, doing whatever they want in a silly way. If they have to die because they don't want to work as part of a team, let them. And don't feel bad. It's easy to stress yourself out thinking you can't let other people die even if they're idiots. That's 100% false. Let them die and then feel confident knowing you're helping them become better players, or helping them not to group with other people and ruining their experience too. (Just don't be one of those silly tanks who throws a strop in dreadsail reef because they didn't get to pull the mobs and so end up letting people die. Those people don't belong in groups).

Honestly? I don't overthink it. I raise all of my characters as DPS until they level all their class skills, get the faction skill lines I want, etc., then I convert them to whatever role I want them to play.

In terms of gear, I don't even bother with making special gear because they level so fast that its not even worth trying to get specific gear. I just grab and wear whatever drops. The only thing I do to "kit" them out is I drop all my xp scrolls on that character and I give them 300k gold to buy their backpack expansions to the max and start training their mount.

In terms of leveling, I just go to the Alik'r desert anchors and just run those until level 10, then I'll do random dungeons to get the bonus XP. Another good way to get easy XP and level up is if you have one of your partners run the character around so you can get wayshrines and discover markers on your map. If the character will need the PVP skill lines, I'll take them into cyrodil and zerg surf until I have those completely unlocked, and often times, I'll get levels (and skill points) that way too.