Brightwing is one of the most powerful supports in the game, in both solo queue and team queue. Before the latest reset I was sitting at 75%-80% winrate with Brightwing with Brightwing being my most played hero by far. Brightwing is one of the few heroes in the game with global presence and has the strongest global presence. Being able to be anywhere on the map within a few seconds to complete turn a fight or save someone from a gank (assuming you're fast enough) will take most opponents off guard, especially disorganized pubs. Brightwing can be fairly spam heavy so you mana management is key... however you can always use your Hearthstone then teleport back into the action. I like pushing a lot of buttons so this happens frequently.

Don't let Brightwing's support tag fool you, either. She can be fairly strong in direct combat and deals a surprising amount of damage while mitigating much of it dealt to herself and allies. That being said I do think that Brightwing pairs best with another support such as Li-Li.
Don't be afraid to wander off and push that lane or take an objective by yourself. Your ability to be anywhere and your fairly high durability will keep you safe or get you to an important objective when needed.

Brightwing is a hero that flows very naturally after a few games. It is hard to put into words exactly how you should be playing Brightwing as that will vary a lot between games. The general idea is to just force yourself to have as much map awareness as possible. Using your teleport as much as possible (in ways that aren't stupid) will be so beneficial.

Your Q can push off small creepwaves and does a surprising amount of damage. Use this as often as possible in team fights.

Your W shuts down key targets and comboed with the talent that makes them take extra damage it will remove them from a teamfight completely with a little coordination. It can also be used to save allies who are being chased.

Your E blocks small amounts of attack damage and increases movement speed. I generally use this to increase movement speed than to block damage because movement speed is real good in HotS.

Both ultimates are solid picks and I slightly prefer Emerald Wind over Blink Heal. Choose based on how the game is going. Emerald Wind is great if you are team fighting on objectives or need to break the base. However, if you are having many smaller team fights on maps such as Cursed with a lot of cliffs than you may want to consider Blink Heal. They are both fairly simple to use as well so just pick what you feel is best. Picking the wrong one or using it incorrectly is far from the worst thing you'll do in any given game.