There are many resources available for mastering individual heroes (like the Cynical Nerds Youtube series or Furious Paul's Overwatch Strategy Guide). Instead of retreading the same ground, I'm going to approach Overwatch's 21 heroes from the perspective of how easy each one is to master. Every hero is fairly easy to pick up and use, but some heroes are more tricky to play in a way that will help your team win a match.
Starter Heroes: These heroes have straightforward, understandable skillsets and roles, and it is very easy to make a difference in a game with them.
Bastion: Bastion's ability to become an immobile chaingun of doom makes him the first hero to impress new players, but he is very counterable by more skilled players. Try to move around to a new position after scoring a particular string of kills in one place, and be aware that you have a "headshot" critical vulnerability on the back of your body in turret mode.
Lucio: Lucio has a passive aura that either heals nearby players or speeds them up. Try to use speed as your default setting, only toggling to heals between battles or when many team members can benefit. You probably won't get many solo kills as Lucio, but you are very good at "boop"ing people off the edge of cliffs:
McCree / Reaper / Soldier 76: These are the traditional damage dealers. They have shooty skills to shoot things, mobility and heal skills for survivability, and ultimates that make them even more shooty. Use them to keep pressure on objectives, support tanks, and protect your healers.
Winston: Winston has good mobility and a weak gun that hits whoever happens to be nearby, making him a good choice for people with poor aim. Use him to provide localized defense in a battle, or to quickly get to a back lines sniper or support for harassment -- he's less useful as a pure frontal assault kind of tank.
Intermediate Heroes: These heroes are simple to play, but require a little extra spatial awareness and focus to support game objectives. Try them out once you're comfortable with the general flow of the game.
Junkrat: Play Junkrat if you like chaos and don't mind being off doing your own thing. Junkrat can shut down a narrow chokepoint singlehandedly with his randomly bouncing grenades, and his kit of traps are fun to play with and useful for staying alive.
Mercy: Mercy is a single-hero healer (where Lucio is a group healer) and can also boost the damage output of a single target. The healing part is easy -- the difficult part is using her "fly to team member" skill to constantly stay ahead of enemies and out of harm's way. Try to constantly be flying between team members rather than picking your favourite to play assistant with. As you improve, healing will become second nature and establishing a good flight pattern will take up most of your time. Mercy is a very fun and rewarding healer to play (She was my second most-played hero in the beta).
Pharah: Pharah has straightforward skills -- fly around and launch rockets! The tricky part here is to manipulate her flying skills in unpredictable patterns. A Pharah flying straight up or down at constant velocity is going to be easy pickings for any sniper. You'll also need to get into the habit of leading your shots to anticipate where your target will be by the time the rocket reaches them.
Reinhardt: Reinhardt is the tankiest of tanks, with a massive shield that's perfect for getting your team through a chokepoint. Don't be afraid to drop the shield and start rage-swinging his giant axe when all of your teammates have seen squirrels and run off somewhere else -- he does a surprising amount of damage and his melee range is big enough to get some satisfying kills on squishy enemies.
Symmetra: Symmetra's gameplay can feel unfulfilling -- give shields to your teammates and plant mini turrets around the map to catch enemies unaware (represented by an icon that looks like a retarded penguin on your screen), in order to use your Teleporter ultimate which gets your respawned team members back to the battle faster. Sometimes it will feel like you're playing your own little minigame within the game. You'll need some map knowledge to identify places where your teleporter won't be found and destroyed by enemies.
Torbjorn: Torbjorn builds a really irritating turret that he can repair over time. Try not to play him in a "protect the turret" mode, as this is very limiting. Instead, let the turret support you as you shoot with your surprisingly painful gun and rebuild the turret for free when the old one falls. Torbjorn is a pretty weak choice when you're on the Attacking team, but I've seen a few that have proved this general rule of thumb to be wrong.
On Friday, I'll cover the other 9 heroes, which are a little trickier to play than these.