RACE

In ESO you can heal on any race. Each race offers something to a healer, but some races offer more than others. The ease of use, the strength of heal, PvE vs PvP, and whether you are a magicka or stamina based healer will influence your race choice.

For beginners I would recommend both of these races as they have strong heals with great resource management:

Argonian - Has a number of racial passives that help healers. Of these, the primary one is the potion passive, which restores 400 health, magicka, and stamina when you drink a potion. They also get 1000 magicka and 6% healing done.
Breton - Has great resource management because of racial passives with magicka recovery, cost reduction to all magicka skills, 2000 max magicka, with extra spell resistances for increased survivability.

For experienced players, for the most healing strength due to the extra damages and magicka pool:

High Elf - With 258 spell damage, 2000 max magicka, as well as 645 magicka or stamina restored after casting class abilities, High Elves just edge out Dark Elves for the strongest healing capability because of the extra 125 magicka.
Dark Elf - With 258 spell damage, as well as 1875 max magicka and stamina, they are only slightly behind High Elves in healing strength. The extra stamina is really nice, though ultimately it really comes down to personal choice.

Stamina Healers are uncommon in ESO, as there are only a small number of healing skills that use stamina. As such, most stamina builds are somewhat a hybrid, using both magicka and stamina abilities:

Dark Elf - 258 Weapon and spell damage and 1875 max staminia and magicka makes the dark elves the number one choice for me by a perceivable mile. As above stamina healers are more hybrid-style build, and with both damage and resource stats in stamina and magicka Dark Elves are really the only class I recommend for stamina healing right now.
Orc - 2000 max stamina, 258 weapon damage, and increased movement buff. If we had a full stamina heal tool kit an Orc would be slightly more powerful than a Dark Elf, but at present it's a distant second due to the lack of stamina based heals.
Khajiit - 10% increase to critical heals and damage, as well as 825 max stamina and magicka. Khajiit are ranked this high because they lend themselves to hybrid builds. The critical healing impacts both stamina and magicka skills and gives a small increase to both max stats.
Imperial - 2000 max stamina and health, and 3% decrease cost to all abilities. Imperials have good sustain, but lack healing power.
Redguard - 2000 max stamina, and restore 950 stamina when you deal direct damage - which can occur every 5 seconds. Redguards have good sustain, but lack healing power.
Wood Elf - 2000 max stamina and 258 stamina recovery. Wood Elves have great sustain, but lack healing power.
Nord - 1500 max stamina and increased ultimate generation while in combat. The increased ultimate generation could have potential in PVP, with increased resistances.