When we get a phone call like this one of our first questions is usually how far can you see down into the pool?
Can you see the bottom?
Can you see the bottom step?
Can you see the second step down from the top?
If there is a lot of algae buildup to the point where it is really dense in the water preventing you from seeing more than one or two feet down, or if the algae is like carpet on the walls then the drain and clean with an acid wash is probably the way to go. If your plaster is in great shape and you have a lot of debris down at the bottom (meaning more than 2-3 large trash bags worth) you may want to do a drain and clean and forgo an acid wash… however… an acid wash will not cost much more in terms of time and chemical cost to perform once you have decided to do a drain and remove all the debris.
If you have a swimming pool that is just turning green or has a green tint to the water but does not have a buildup of algae on surfaces and does not have a lot of debris in it then I would say shocking it and performing algae treatments via chemicals is typically the way to go.