If you mix classes does one fighting ability diminish the other? Say I chose a barbarian and a hunter, would the fighting styles change, do I have to chose one, or does nothing happen to both styles?
When you multiclass, (see multiclassing rules), there are a few notable trade-offs.
Generally speaking, all of your features will add together, however, there are some which will not. For example, the Extra Attack from Ranger does not add to the Extra Attack from the Barbarian (or any other class), so splitting a class Ranger5/Barbarian5 wastes an important feature. Fighters get an enhanced Extra Attack, which stacks with itself, but not similar features from other classes.
With spellcasting classes, such as Wizard5/Cleric5, you continue to gain Spell Slots as though you were Wizard10 or Cleric10, however, you can only learn spells accessible by a Wizard 5 and Cleric 5, so your low level spells are more powerful, but you delay access to higher level spells.
Warlocks have Pact Magic, which is entirely its own thing, so while multi-classing Warlock/Sorcerer is common, you track Pact magic and Spell Casting separately.
Beyond that, you generally can't stack benefits from identical sources, so you wouldn't be able to choose Fighting Style: Archery from two classes. You would need to choose different styles.
Short answer: no, multiclassing does not change the features of the other class, but features with the same name usually have extra rules as to how they combine (or don't).
Also, some multiclasses work better together than others. In your example of a barbarian / hunter ranger ... rangers can cast spells like hunter's mark that increase the damage of their attacks but the spell requires concentration which doesn't work with the barbarian rage feature. So a raging barbarian may have trouble using some of the ranger spells in combat.
If you mix classes does one fighting ability diminish the other? Say I chose a barbarian and a hunter, would the fighting styles change, do I have to chose one, or does nothing happen to both styles?
When you multiclass, (see multiclassing rules), there are a few notable trade-offs.
Generally speaking, all of your features will add together, however, there are some which will not. For example, the Extra Attack from Ranger does not add to the Extra Attack from the Barbarian (or any other class), so splitting a class Ranger5/Barbarian5 wastes an important feature. Fighters get an enhanced Extra Attack, which stacks with itself, but not similar features from other classes.
With spellcasting classes, such as Wizard5/Cleric5, you continue to gain Spell Slots as though you were Wizard10 or Cleric10, however, you can only learn spells accessible by a Wizard 5 and Cleric 5, so your low level spells are more powerful, but you delay access to higher level spells.
Warlocks have Pact Magic, which is entirely its own thing, so while multi-classing Warlock/Sorcerer is common, you track Pact magic and Spell Casting separately.
Beyond that, you generally can't stack benefits from identical sources, so you wouldn't be able to choose Fighting Style: Archery from two classes. You would need to choose different styles.
Memnosyne pretty much covered it.
Short answer: no, multiclassing does not change the features of the other class, but features with the same name usually have extra rules as to how they combine (or don't).
Also, some multiclasses work better together than others. In your example of a barbarian / hunter ranger ... rangers can cast spells like hunter's mark that increase the damage of their attacks but the spell requires concentration which doesn't work with the barbarian rage feature. So a raging barbarian may have trouble using some of the ranger spells in combat.
Thanks