Because this will be my first campaign, I decided to share the house rules I'll be using with all of you. I've decided to keep this list pretty short, wanting to run most things by the book for the time being. I want to keep things simple for both my players and myself for the time being. I'll most likely update this list in the future if I make any additions or cut something from the list.
HOUSE RULESAbility Score Generation
Players will generate their character's ability scores by rolling 4d6, dropping the die with the lowest result, and adding the remaining three together. They'll repeat this until they have six numbers, then assign them to their abilities as they see fit.
Hit Point Generation
Characters will begin with maximum hit points at 1st level, plus their Constitution modifier. Every level thereafter, the character receives additional hit points by rolling their hit die twice, keeping the die with the higher result, and adding their Constitution modifier.
Players can select any non-evil alignment for their characters without needing Dungeon Master permission.
Healer's Kit Dependency
Characters can't spend any Hit Dice after finishing a short rest until someone expends one use of a healer's kit to bandage and treat the character's wounds. This variant rule can be found on pg. 266 of the Dungeon Master's Guide.
Cleaving Through Creatures
When a melee attack reduces an undamaged creature to 0 hit points, any excess damage from that attack might carry over to another creature nearby. The attacker targets another creature within reach and, if the original attack roll can hit it, applies any remaining damage to it. If that creature was undamaged and is likewise reduced to 0 hit points, repeat this process, carrying over the remaining damage until there are no valid targets, or until the damage carried over fails to reduce an undamaged creature to 0 hit points. This variant rule can be found on pg. 272 of the Dungeon Master's Guide.
When a character receives an unmodified result of 1 on an attack roll, they have critically failed. This fumbled attack gives the next attack roll made against them advantage.
Identifying a Magic Item
A character can focus on one magic item during a long rest, while being in physical contact with the item. At the end of the rest, the character learns the item's properties, as well as how to use them. This rule replaces the default rule for identifying a magic item.
Feel free to leave any comments about the house rules presented above. I tried to keep things short and simple, but I will definitely be evolving this list as the campaign goes forward. Go ahead and share some of your own 5th Edition house rules in the comments below. I'd love to see how other people have tinkered with the mechanics to personalize their own experiences with it.