For example, if a goblin tries to ambush the party, and one round later, another goblin tries to ambush the party. If a character wasn't surprised at the first goblin but got surprised at the second, and the character already did their turn, would they not be able to take an action in the third round?

Unless the second goblin can't surprise the party, which doesn't make sense 

Edit: Oh wait, found it

Sage Advice

Surprise rules work for two opposing sides. What happens with surprise when a third group of combatants sneaks up, hidden from the melee, and ambushes?

The surprise rule is relevant only when a combat is starting. Any ambushes during the fight use the rules for Dexterity (Stealth) checks.

Yup, surprise would better be named as flatfooted.  Once people know there's a fight going on, they're prepared to react to changes in circumstance.

Surprise can only happen in the first round of combat.

A monster can attack from a hidden position as part of its attack, so the second goblin would have advantage on its attack in the second round (assuming it remained hidden, and wasn't seen, in the first round).

Hidden =/= Surprise.

A hidden creature (stealth check exceeds the passive perception of the creature it's attacking) will have advantage on attack rolls against that creature. This can occur at any point in combat.

Surprise applies only to the first round of combat.