Customer Engagement appeared in the marketing lingo in the early 2000s, along with the concept of Touchpoints. In some ways, CE has become the Holy Grail, replacing such simple measures as Customer Satisfaction (which only looks at one point in time) or even Customer Loyalty (which doesn't have the emotional component.)
Gallup introduced an 11 question metric for CE back in 2001 which includes 3 measures of loyalty (overall satisfaction, intent to repurchase, intent to recommend) and 4 overall emotional attachment factors:
CONFIDENCE : always trust X, X always delivers what they promise
BELIEF IN BRAND INTEGRITY: X always treats me fairly, X can be counted on to reach a fair and satisfactory resolution of problems
PRIDE: proud to be a customer, X always treats me with respect
PASSION: X is the perfect company for people like me, I can't imagine a world without X
Back to the Journey - this is then the stages that a customer travels through, typically something like awareness - consideration - inquiry - purchase - retention. And there are plenty of visual depictions of this ranging with funnels, arrows, boxes, cycles and all the other delights to be found on Powerpoint.
Just two words of warning here. The idea of "stages" and names that echo old communication models such as AIDA do tend to suggest a linear and rational process. While it may be helpful to try and visualise the Journey, it's always remembering that "the map is not the territory", as Captain Scott and others found out to their cost.
And however much we may try and rationalise this into a process, human decision-making is not rational. Think back to the last time you bought, say, a pair of jeans. Did you draw up a nice "consideration set" with a table of prices and availability? Thought not.