If I have a lot of 1/1 Goblins on the battlefield and my opponent casts Plague Engineer can I save them all by responding with Fatal Push on the Plague Engineer or does the -1/-1 effect occur before I can kill Plague Engineer? Sorry, I seem to be struggling with a very basic rule here.
From the ruling on Gatherer for Plague Engineer:
The choice of creature type is made as Plague Engineer enters the battlefield. Players can’t respond to this choice. The -1/-1 effect starts applying immediately.
The reason this is the case is because plague engineer's ability is static. You can't respond to the choice of creature type because it's not an ETB trigger. For as long as the engineer exists on the battlefield his ability is in effect, and he needs to be on the battlefield in order to target him, so once you are able to target him your goblins have been dead for a while.
To go a little further with ApotheosisCM's answer, the effect on Plague Engineer
As Plague Engineer enters the battlefield, choose a creature type.
is a replacement effect, which means that, instead of the creature entering the battlefield normally, you take an additional step as he enters the battlefield i.e. choose "Goblin". The critical thing to note is that this does not use the stack, it just takes place as a part of the Plague Engineer resolving.
Replacement effects can generally be distinguished from triggered abilities due to the use of the word "As" at the beginning of the effect. If the ability was instead worded as
When Plague Engineer enters the battlefield, choose a creature type.
Then it would be a triggered ability, and you would have a chance to respond with Fatal Push. However, I don't know if there are any cards that are phrased this way.
Unfortunately not. In order for you to target Plague Engineer with your Fatal Push it has to already be on the battlefield. That means that it's effect has already occured and your gobbos are gone. If you had countered it with a Counterspell then it wouldn't hit the battlefield and your 1/1s would be safe.