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Our adroit heroes tracked the goblins back to a small camp in the woods. Completely unaware of the threat around them, the goblins sit around a fire, easy prey to the hidden band of adventurers lurking in the trees.

The party attacks! Arrows rain down, spells are cast, and sword wielders rush in. Our party...

Attacks with advantage because they are hidden?

Only gets a surprise round for catching the goblins unaware?

Both get surprise and advantage?

What are scenarios in which each of the above would occur?

It's all in your description:

Completely unaware of the threat around them, the goblins sit around a fire, easy prey to the hidden band of adventurers lurking in the trees.

You, the GM, have already determined that the goblins are completely unaware and that the adventurers are hidden. (Some GMs might have compared stealth vs. perception, some might let the fiction do the talking for them, you may have done neither; see DMG p. 237 for more on how to decide whether or not to require checks in this instance.)

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  That the goblins are completely unaware means they are surprised. From PHB p.189 at "Surprise":

  
  
  

  
  

  Any character or monster that doesn't notice a threat is surprised.

  
  

  
  


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  That the adventurers are hidden means that they attack at advantage. From PHB p.195 at "Unseen Attackers and Targets":

  
  
  

  
  

  When a creature can't see you, you have advantage on attack rolls against it.

  
  

  
  
  

  So as you describe it your archers and (presumably-ranged casters) would have advantage. Those sword-wielders, though... they lose their advantage as they come into sight and attack (presumably while visible) the goblins in melee.