Tyrande. The Healer new accounts can pick as their first Hero, and like Jaina she's given to you for free if you bought Spoils of War. I personally think she probably shouldn't be the introductory Healer, as she's very dependent on good use of continuous stutter-stepping to enable her healing, Lunar Flare is generally at its best when coordinating with a teammate, and Sentinel relies on a good awareness of the game state to leverage its scouting utility and potential for long-range executions, all of which are not liable to come readily to a newbie; almost every other Healer is more newbie-friendly. Tyrande herself is good at enabling executions of key targets and foiling sneaky enemy tactics.

Li Li. A Healer with some design elements obviously aiming for accessibility to new players, and in practice one of the most skill-demanding Healers to bring out her full potential. Also one of the deadliest duelists out of Healers, which pretty clearly often catches people off guard.
Kharazim. A Healer who behaves more like a highly mobile Melee Assassin with healing as a side-gig, especially if you don't take Transcendence at Level 1.
Malfurion. A Healer who excels in big teamfights, especially against enemy teams that like to cluster, and able to empower Mana-using allies, cut off escapes or pursuits, all backed by surprisingly decent damage.
Alexstraza. A Healer whose effectiveness drops if she's focused by the enemy team and rises if she's paired with high-HP teammates, especially if her team is low-mobility and likes to cluster. Also frustratingly vulnerable to enemy tools that punish clustering.
Uther. A Tank-like Healer who can ward against dives, substantially ablate burst damage if timing his heals well, and prevent enemy escapes, but with poor healing throughput per se.
Auriel. A Healer who can stay in the field indefinitely so long as she has a partner with high damage output, can punish unwary dive attempts, and once her Heroic comes online either reliably prevent or undo a teammate's death. (Or even her own death)
Deckard Cain. A Healer who is particularly great if he arrives early and preps the field for his allies, but with very good healing throughput in general and deceptively high durability and good ability to pin enemies down so his allies can finish them. The only Healer in the game where being dead doesn't necessarily stop their healing.
Lieutenant Morales. A Healer with incredible ability to keep her team alive indefinitely, especially against attempts to focus a single target... so long as that target isn't her.
Stukov. A Healer who excels in teamfights, especially against teams that prefer to spread damage around, with significant long-range disruption and once he has a Heroic he also has great ability to keep enemies off of him. Also surprisingly lethal in-close.
Whitemane. A Healer who directly translates damage to enemy Heroes into healing for her team, who thus appreciates allies who can enhance her damage (Such as by lowering enemy Armor) and can struggle if the enemy team is good at reducing her damage.
Lúcio. A fast and mobile Healer who can get his entire team moving fast or healing when they need it, run down fleeing foes and prevent their escape, or charge in and enable an ally's escape.
Anduin. A well-rounded Healer with the unique ability to pull allies to safety. Garrosh's best friend.
Rehgar. A fast and hard-hitting melee Healer, good at escapes and pursuits, and with the innate ability to protect allies from disables.
Brightwing. A Healer whose presence passively heals her allies if they stick close to her, with the ability to heal and reinforce an allied Hero anywhere on the map periodically, and with impressive ability to disable/disrupt enemy action. Not very combat-mobile by default.
Ana. A Healer who heals allies from a safe distance, can strategically deny enemy healing on key targets, and can shut down dive attempts so her teammates can punish them. Pretty bad at keeping her own HP up, though.