Alerts are indeed instances. Queue up for them and you'll run them with a group. They're sorta like wow dungeons, but tend to be more open.
"Legends" are a type of pvp where you play as an iconic DC hero. You should have gotten a legend chip in the mail, use it to add that legend character to your list of choosable characters. If you're a hero, it'll be robin. If you're a villain, it'll be harley. All other legend characters have to be purchased using the game's version of honor points.
One thing to note about legends pvp. No matter what level you are, the iconic character you take control of will be max level. So your own gear and abilities don't matter in legends. You will need to practice with your chosen character's abilities though, since they each have their own loadouts already pre-chosen.
Also, at level 10, you get the ability to change stances. You've been in DPS stance the entire game. Now you can switch into whatever stance your class has. Tank mode for tanks, healer mode for healers, or controller mode for, well, controllers! You'll do less damage (especially if you're healer) but you'll be better at your other role.
If you're a controller, I highly recommend giving controller stance a try. Controllers don't get a damage penalty like the other classes (though you'll still do less damage because you won't have the damage bonuses of dps stance) -but- controller mode gives you a ton of power regen. A TON OF IT. You'll be so full of power you won't know what to do with it.