Rules are here.
- 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.
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).