My group and I are very new to D&D so I need help! The adventures were inside a hideout. They came up a closed door and heard goblin noises inside. The fighter decided to break down the door and rush in. The rules say that surprise only occurs when one party is being stealthy (which he wasn't doing because he broke down the door) so the surpise round wouldn't happen. They would not get automatic surpise right?...if not, would the fighter get an advantage on his turn? We were thinking that the fighter would get 1st in initiative order and maybe get an advantage on attack (or all adventures that are in initiative order before the goblins get advantages).

Both things are ultimately adjudicated by the DM, but Surprise is often granted when one group catches the other flat-footed; I would say if your Fighter was able to break down the door in a single moment rather than pounding on it long enough for the goblins to become alerted and ready themselves and they weren't already on guard, they'd be Surprised, which means they won't act on the first round of combat. Advantage in the context of combat is probably best left tied to conditions and features; catching opponents by surprise does not innately confer advantage.

Now, if say this was a combat encounter being kicked off by a character shooting from the shadows before the rest of the party charged an off-guard group of foes, the initial attack would have Advantage by virtue of being unseen while the enemies would be Surprised and thus miss the first round of actions.

If the goblins were unaware of the Fighter’s presence prior to the door smashing, I would personally have said the Fighter surprised the goblins and they (and the other PCs) would not have acted on the first round of combat due to their being surprised.

RAW, you would have rolled initiative and determined surprise (stealth checks vs passive perception) before breaking down the door, then the fighter would break the door on its turn.

I personally don't care for RAW surprise.

Quote from DxJxC >>

RAW, you would have rolled initiative and determined surprise (stealth checks vs passive perception) before breaking down the door, then the fighter would break the door on its turn.

I personally don't care for RAW surprise.

The only think I don't like in the RAW surprise rules is "If neither side tries to be stealthy, they automatically notice each other. " [and the implication that this means noone is surprised].

Depending on the door and how much noise the goblins are making it could be reasonable to assume the goblins are unable to hear the party approach even if they are not trying to be stealthy. If a changling disguised as a friendly NPC suddenly attacks the party they should at least have a chance of getting surprise even though they are not being stealthy (perhaps deception vs passive insight)

In the above case RAW the party members who are ahead of the fighter in the can hold their actions. The Goblins who beat the fighter on inititave have quick enough reactions to respond to the attacks (for example if the fighter breaks down the door and the rogue has a held action to fire an arrow at the goblin wizard. If the goblin wizard has the shield spell where they are on the initiative order determines whether they are quick enough to cast shield against that attack)

Quote from DxJxC >>

RAW, you would have rolled initiative and determined surprise (stealth checks vs passive perception) before breaking down the door, then the fighter would break the door on its turn.

I personally don't care for RAW surprise.

 
You could theoretically compare passive perception to passive stealth too.