Path of the Phoenix

The Phoenix Path is used for concussive AOE damage and being able to dart around the battlefield, as well as giving the barbarian much deserved range in a fight. This path also gives you more variation beyond what I listed before, if you take anything away from this class is that it is a multi-tool that gives barbarians more than RAGE->ATTACK repeat. 

Pyro-Rage

Starting when you choose this path at 3rd level, you can go into a rage you burst with primordial fire dealing 2d6 in a 15ft radius, this increases to 3d6 at 6th 4d6 at 12th and increases the radius to 30ft at 15th feet. Your weapon attacks also streak with fire dealing an additional 1d4 fire damage, this increases to 1d6 at 14th level. When your rage ends, you suffer one level of exhaustion.

On Fires Wings

Beginning at 6th level, while raging you can use your bonus action to dash towards an enemy and impact into them dealing your Pyro-Burst damage. At 10th level you can use this to go up vertically or downwards if in midair. (the dash is your full movement 30ft or whatever increases you may have.)

Soul of Fire

Beginning at 10th level, you gain resistance to fire damage and as an action you may point towards a target and cast the Fire Ball spell, this ability recharges on long rest. Also if you fall in combat, (going to 0 HP) you burst into fire dealing your Pyro-rage damage and you go to 1 HP. (if you are a half orc or have another ability like the Half-Orc racial ability that triggers on 0 HP, you may use that first and this one second.)

Phoenix Aura

Starting at 14th level,  when you rage you emit a violent heat, any creature that ends their turn within 15ft takes 1d6 fire damage, this increases 1d6 per consistent round to a max of 3d6 per round. (Example: 1st round takes 1d6 2nd: 2d6 3rd: 3d6 4th:3d6) You may also get a running start and jump 60ft feet using fire to boost you in any direction (This may stack with On Fires Wings).