Generally the order of reactivity of alcohols follows the order tertiary > secondary > primary. But why is it reverse in case of esterification.
Do you know the mechanism of esterification? The first step - which is also the rate determining step - involves the nucleophilic attack of the lone pair on the alcohol's oxygen atom on the electrophilic carbon atom of the carboxylic acid. Hence, the more nucleophilic the oxygen atom, more will the equilibrium shift to the right, favoring esterification.
Now, can you tell which degree alcohol is the most nucleophilic? And can you tell now which degree alcohol will favor esterification the most?
For Lucas reagent, the reaction involves the formation of a carbocation (at the position where the hydroxyl group originally was) by elimination of the hydroxyl group as an oxonium ion. Thus, the more stable the carbocation, the more the Lucas reagent favor the white precipitate.