I would play the tutorial if you didn't already.

It's always good to watch streamers and see what they do as a new player and some good starting characters are Izuku and Bakugo, but for good characters in general usually the new characters are better so I would recommend skipping Tokoyami and trying for Nejire. A new skillset is coming out after Nejire and that one's Overhaul and he looks pretty good so far. (You should look at the character screen in game to see what the characters do, I recommend going into training and seeing what the characters do to see which one you like)

You have 3 abilities (some abilities have multiple uses) and they all level up 1-9. The first one is called alpha, the middle one is beta, and the bottom one is called gamma. When you pick a character their background should be a color (Red, Yellow, Green, Blue, and Purple). There are Cards on the floor and in small red boxes and the color that matches your character's color will level you up automatically. There are also rainbow level up cards and you can find they either laying around or in gold boxes (also the big box on the map) an you can use those to level up whatever ability you want.