When trying to show which one to apply, how do we know which one we should use? What does one specifically look for?
Delta method uses the CLT, so you'll necessarily have a convergence to a normal distribution. Slustky is for a convergence in distribution, more general.
If you're given a rv that converges to a constant, then you'd have to think of Slutsky.
For the delta method, it's often used for estimators of parameters...
You can have a look at the wikipedia for more information