In our consumerism-driven culture, these three words "savour", "cherish", "treasure" seem to have little or no space in people's lifestyles. We are so accustomed to "consuming" something--be it a thing, an event, a person, an experience--and moving so thoughtlessly and unmindfully to "the next thing" that could titillate our consumeristic instincts with something like a new "high."
In an age or a lifestyle that had/has fewer "things" and fewer "stimulants," someone could in principle "savour," "cherish," and "treasure" each and every experience--both positive and negative--better, learn what needs to be learned and emerge from it a richer, more whole, more mature person.
This is why these three words could potentially be some of the most important keywords of a lifestyle that is necessary, especially today in our crazy consumeristic and frenetic world.