King of Tokyo: When can you buy a card and when can you start to use them?
Rules are here.

Purchasing Cards:


  Turn Overview
  
  
  Rolling and rerolling the dice
  Resolving the dice
  Buying cards (optional)
  End of your turn
  


So you can only buy cards during your turn, and only after rolling and resolving the dice. One exception to this is if you have the Opportunist card, in which case you can buy newly revealed cards even when it's not your turn.

Using Cards:

Cards marked Discard


  Monsters immediately resolve these then discard them.


Cards marked Keep


  Monsters keep these cards for the remainder of the game (sometimes the card will indicate it is discarded under certain circumstances.)


Keep cards generally indicate on them when they can be used (e.g. when attacking, when you are attacked, at the start of your turn). A Keep card that does something at the end of your turn can be used on the turn that you bought it (e.g. Solar Powered).