Pyromancer Sorcerer

I'm admittedly surprised that D&D hasn't created a Pyromancer subclass for the Sorcerer.

While Dragonblood Sorcerers can tap into fire powers, it's not the same as being a true pyromancer. In my mind, Pyromancers would have similar characteristics to Storm Sorcery, but with fire. I'm not sure what kind of effect they'd have should they cast a level one spell or higher, like how Storm Sorcerers gain the ability to fly without triggering opportunity attacks. Maybe, everything in a five foot radius from the caster takes fire damage, feeling the heat coming off the spell, and have to make a Con or Dex save in order to take half the damage. Or maybe that could be saved at higher levels, similar to how when a Storm Sorcerer takes damage by a melee attack, so does the attacker. And of course, they could gain resistances to fire and cold damage at Level 6 and then gain immunity to fire at Level 18.

This should probably be in the Homebrew forum tbh

You basically described most of the features from the Pyromancer subclass from the free PDF "Plane Shift: Kaladesh", except you missed the part where you can break through resistance and immunity.

Granted, that was playtest material created more by the Magic: the Gathering team, not official content printed for use, but you could absolutely insert the subclass as a homebrew one and use it with your DM's approval. Just remember to adjust the level 1 feature Heart of Fire to level 3 if playing 5.5e.