Here's how you do it:
1. Pick class.
2. Get resto staff.
3. ?????
4. Profit!

But really, and class can become a support/heals character. My one piece of advice to you is to also try and mix some other roles into your build. ESO is all about focused flexibility, and if you go "ALL HEALS ONLY HEALS ALL THE TIME" then there's a good chance that you'll run into some frustrating moments.

I play a templar healer, and don't have any problems with mana. (Course, I spec into mana almost every level, and never touch stamina. And I wear light armor, ofc.) Sure, you can play other classes and still be an effective healer, but templar has some nice skills down the Resorting Light (healing) branch. The passives there also make for really good support... Things like Master Ritualist, which lets you res. faster, return more health on res. and potentially regain a soul gem.

So, yeah, templar is my suggestion if you want a pure healer. However, like @SirPuppingtonVonHat said, going pure healer has serious drawbacks. You won't be able to fight on your own very well. So if you're ever playing the char when your friend's offline, you'll want to party up with someone nearby. (IMO, that's part of the fun of being pure healer, you'll always be a valued group member.) Finally, go Argonian if you really want to be the optimal healer, since their racial perks are complementary.

The truth is, there is no best healer. All classes can heal but only the Templar can heal without a restoration staff. And despite everything you've read Templar's are very fine healers in light armor. http://tamrielfoundry.com/forum/classes/class-discussion/

Take a look here for further info.