First there are a few questions that need to be answered.
Are they wild, holdover or frshly stocked fish?
Can you see fish actively feeding to get an idea on where to fish in the water column?
Is the angling pressure high?
What types of food are available to the fish.
These major things need to be addressed when standing on a river for the first time.
As for techniques, I like to nymph fish so that would be my primary approach.
However, if the trout were extremely active I might choose wet flies.
If the water is high and dirty I might streamer fish or nymph with attractor type patterns.
I would not be concerned with hopping from spot to spot as I would be focusing on fishing one area really well. This will tell you the type of water that the trout there prefer. This might aid you in finding fish in other sections. For example, If the river gets pounded you might fish the "good" spots and only get a few. However, if you fish secondary locations you might find lots of eager trout. Don't try to cherry pick when visiting new streams. Spend lots of time covering one area and you will learn a lot about the new waterway.