Overwatch is Blizzard's take on a team shooter, much like that old classic Team Fortress 2, but way more accessible and with less pointless hats. If you've just picked up this game and have no idea where to start, then this guide will point you in the right direction. Get out there and bust some heads, champ!
If you're new to team shooters in general and you only learn one thing before going in, then for the love of all you hold dear, let it be this. Overwatch had four support heroes at release, and may or may not get more down the line, and if you see one then that person is the person you need to kill. This goes double for Mercy, the mecha–angel with a staff, who has the most immediate healing powers as well as being able to buff damage, which is an upsetting combination. Most of the support characters also have an ultimate ability that will do a mass resurrection/heal/shield on the team, so basically they just need to be off the field as quickly as possible. Take down the support and the rest will follow.
There are many good reasons why a player can't or won't use voice chat, but if you can, you should. While Overwatch is fast moving enough that you can happily play in silence, and has in game compensation (e.g. characters on your team will tell you to watch your back if an enemy is shooting from behind) it's also easier if even one player has their mic up. You can call out enemy movements to one another, let others know where you're going, and warn people if you get taken out by a hidden gun turret or a sniper in a window.
Overwatch has a big roster of characters that can seem intimidating. Soldier 76 is a good starter character, because he does decent damage and can heal himself, so you can use him to get a feel for the game, before learning some of the other characters. Lucio is another great one, because his healing effect doesn't require you to really be involved, and you can still do damage and help your team out.
Sometimes you'll be playing a mode or a map where you don't need to do much other than gank on the other team, and that's fine, but sometimes you need to actually think about what you're doing. If you're playing attack on an Escort mission, for example, and none of your team are actually attempting to move the payload, then you're not winning that match. In a Control game the more of your team are in the control point the faster you can capture it. And so on and so forth.
As in any online game the players who have a greater knowledge of the map will have the advantage. You can predict what movements the enemy team will make before they make them, set up hidden traps and turrets where they won't look for them, and generally walk all over their unprepared asses. This will take time, though, so don't be too discouraged if someone else is currently doing that to you. Your time will come.
This is something Blizzard has been emphasising from the start, but part of being successful in Overwatch is reacting to the other team's tactics. If a Reinhardt is giving you trouble try switching to Pharah, who can boost over his shield to flank him, or set up as Bastion to quickly take the shield out altogether. If fast movers like Tracer or Lúcio are giving you trouble then use Mei's weapons to slow them down. As you play more you'll learn which characters are best to go up against others.
It's tempting to immediately run back into the fray after you've respawned, but coordinated attacks as a group work better. There's not much point to a tank drawing fire if you're not drawing fire from anyone, and support characters and builders often need someone watching their back. Plus there's always the point that if you charge in or go out on a limb on your own you're more likely to be killed. It's a team shooter, so try and stay with the team.
The nature of team shooters is that they can be glorious one moment and hellish frustration the next, because you won't know how good the team you're on actually is. But there is entertainment to be found even if you're on a team that's a burning hot mess, and you can watch the tactics the other team used. Figure out how they beat you, then take that knowledge and use it against the next chump. It's a beautiful cycle.