How long should it take concrete to fully cure?
Concrete will never fully cure; it continually hardens forever, although for all practical purposes, it reaches a point where further hardening will be so slow as to unnoticeable. 

Unless the mix has retarder (or some cement replacement material) in it, it will generally be hard enough to walk on in less than 24 hours. 

Mixes are usually specified with a design strength - the compressive strength that the concrete will reach in a period of time, typically 28 days. As a general rule, at 7 days it will have reached around two thirds to three quarters of the design strength.

Don't worry about the water on the new concrete - it will help the concrete cure if it is kept damp after the initial set. The early stages of concrete curing is the hydration of the cement in the concrete.