Salt and water don't go bang! Why not?
 

A favorite chemistry demo is the tapping of a freshly cut slice of the metal sodium into a bowl of water to show it fiizzing about the surface as it reacts exothermically with the water.

Yet add sodium to water when it is combined with chlorine, in the form of sodium chloride, and it just quietly dissolves as the sodium and chlorine ions dissociate. Why is this? I assume it must be something to do with the rearrangement of the electons...

correct, the sodium and chlorine have already reacted with each other and the sodium has lost an electron to the chlorine. Both now have electron structures like a noble gas nd as such are unreactive.
A "reaction" can be measured however adding salt to water will (i think) cool it (i cant remember off the top of my head) this is because enery is needed to break the rystals of salt down. THis energy comes from the water.