Beat Monsters

These are called Rhythm Monsters in most languages that aren't English. But we're special.

Outline
Put a Beat Monster on a team. Play a song. When the note gets close to the Beat Monster, tap in its lane to get points.

When playing songs one at a time, match your beat monsters' colour with the song colour for maximum points.

When playing multiple songs at once (in events), choose a team with the best overall PR as the song colours will be unknown (other than the first song). This way, you'll receive decent scores for any song colour that shows up.

Rarities
Monsters come in different rarities. The rarer they are, the more powerful they are, in general.

0-star
These are referred to as Common. Their stats are generally poor and you want to get 4 of each color, then replace them with Uncommon monsters as soon as you can.

They come in all six colors, 2 varieties to a color, for 12 in total. The Blue Joker is particularly important for being donation material in the Event Zone.

You can evolve them once. I've never taken the effort to do this, so maybe an evolved Common is game-breakingly powerful. But I doubt it.

Find these in the Main Story, Event Zone (sometimes), Missions, Beat TV or Story Grab.

1-star
These are referred to as Uncommon, despite the fact that they're pretty common. Their stats are still bad. Replace them with Rares as soon as you're not using Common monsters in your team.

They come in all six colors, but there is only 1 variety to a color.

You can evolve them once, but the resulting Beat Monster is weaker in its own color than essentially every 2 star of that color, and most 4 stars of any color.

Find these in the Main Story, Event Zone (sometimes), Missions or Story Grab.

Special
These are used to evolve monsters 2-star and above. There are 2 categories of Special Beat Monster.

Colored Special Beat Monsters
These come in all six colors, one variety to a color They are used for evolving 2-star Monsters (1), and unevolved 3-star Monsters (4).
 * Larvolt, Tao, Wisp, Spark, Pyromite, Buzzer (G,B,Pu,Pi,O,Y)

Get these in events, the Special Beat Monster Generator (1 / 2 days) or occasionally the Story Grab.

Rarity Special Beat Monsters
Someone should add how to evolve evolved 4-star and 5-star Monsters, as I have no idea how.
 * Supra is yellow. Use 4 of these to evolve evolved 3-star Monsters.
 * Ultrix is green. Use 4 of these to evolve unevolved 4-star Monsters.
 * Epika is orange. Use 4 of these to evolve unevolved 5-star Monsters.
 * Legenda is purple. I have no idea what to do with these.

You can get these from the Essence Trader for Essence of the rarity corresponding to their name.

2-star
These are referred to as Rare. Their stats are not terrible, but replace them with Supers as soon as you're not using Uncommons or Commons in your team.

They come in all six colors.

Some 2-stars can presumably only be found with Story Grabs at a very low probability. I have no idea where else you'd get them. But anyway, I'd like to pretend they don't exist. The ones that can be found in the story are far more common, and there are 12 of those.

You can evolve these, and all above rarities, twice.

Find these in the Main Story, Event Zone (sometimes), Missions, Beat TV or Story Grab.

3-star
These are referred to as Super. Their stats are OK. A team full of these might be competitive in the Arena until maybe stage 7.

Find these in the Main Story, Event Zone (sometimes), from the Essence Trader, Missions, Premium Grabs, Featured Grabs, Secret Grabs, or Story Grabs. Very low probability for anything except Premium/Featured Grabs, though.

Twice evolved Rares are also Super. They will be found in Premium and Featured Grabs as well. Until they get to a very high level, they are weaker than base Supers, so generally, it is not advisable to use them if you can fill your team with base supers.

4-star
These are referred to as Ultra. Their stats are good. A team full of these is competitive at Tier 4 or 5 in the Arena.

The Essence Trader is the most reliable way to get these, but they can also be found in the Premium, Featured and Secret Grabs.

Twice evolved Supers are also Ultra. I know nothing about them, as I do not have any.

5-star
These are referred to as Epic. Their stats are great. A team full of these makes you one of the strongest people in the game at the moment.

The Essence Trader is the most reliable way to get these, but they can also be found in the Premium, Featured and Secret Grabs.

Base Epics are nice because they are stronger than any Super, even those of a different color. This means that they can replace a Super in a team.

Twice evolved Ultras are also Epic. I know nothing about them, as I do not have any.

6-star
These are referred to as Legendary. No one has these, but they definitely exist.

Twice evolved Epics are Legendary. I do not know how else you would get one.

7-star
These are referred to as Mythical. They may or may not exist.