While she's listed as a support, it's important to know that Overwatch's Symmetra really isn't a replacement for Mercy or Lucio. She has some incredibly useful abilities but healing heroes is not her thing. What is her thing is setting up deadly traps with her Sentry Turrets and using her ultimate to get heroes back into the fight instantly. If you like being a support but aren't keen on healing, Symmetra is an interesting choice.
Photon Projector [LMB/RMB] - Symmetra's main weapon is also her primary way of farming up enough energy for her ultimate, so using it often is a necessity. The LMB fire unleashes a stream of energy that does damage that increases the longer the beam is damaging a target. The alternative fire mode charges up slow-moving orbs that can deal considerable damage if they land—but enemies will see them coming a mile away.
Photon Shield [E] - Symmetra grants allies a rechargeable shield that remains until they die. A pretty simple ability with a short cooldown that means you'll always want to make sure everyone on your team has one.
Sentry Turret [LSHIFT] - Symmetra can place up to three of these small turrets along any surface that fire a laser at anything that gets too close. The damage they deal can be quite deadly to squishier targets, but they're hardly a substitute for Torbjorn's turret or Bastion. Placing them against walls where enemies are likely to move past them and not notice is ideal, as they're best used to help up your team's damage-per-second.
Teleporter [Q] - The real reason that Symmetra is a hero worth choosing all comes down to her Teleporter. With this, she can place a portal that opens a link between your spawn point, letting you and your team get back into the fight really quick. While it's active, the enemy will get alerts to go searching for it, so you'll need to hide and defend it properly. Only six allies can pass through it before it collapses.
Symmetra Strategy Tips
As mentioned above, Symmetra isn't capable of being a support for a team that requires healing. Because of this, knowing when to choose Symmetra will require a good knowledge of other heroes or requiring someone else to play a Mercy or Lucio as well.
Another tricky thing about her is her Sentry Turrets, which are really only useful on maps where Symmetra is defending. They die extremely quickly and don't pose much of a threat. Place them in areas where a Bastion or another hero will likely draw the attention away from them, letting the Sentry Turrets contribute without being blown up.
Really, Symmetra boils down to her extremely useful ultimate, which can keep momentum going on a push but isn't as useful when defending. When attacking, the portal can significantly cut down on the travel time to the objective, but defending the final objective on most maps gives the defending team an advantage in the form of a short walk back into the fight, making the Teleporter all but useless.
Placing the Teleporter is going to be tough since placing it too far back will keep it safe but not help your team and putting it too close to the frontlines risks having it destroyed. Either way, the enemy team is going to know when it's active and, if they're smart, a Tracer or Genji will come looking for it. In that case, it's possible to use the Teleporter to set up elaborate traps when used with Sentry Turrets to bait out enemy heroes and kill them.
Farming that Teleporter will take a good deal of skill, however. Symmetra's Photon Projector takes time before reaching maximum damage, and you'll need to get really close to actually hit an enemy with it. Because of this, staying alive as Symmetra is tough, making her an unlikely pick for a lot of players.