Revenge may well be "a dish best served cold" (as the movie Kill Bill! would put it). However, it rarely is. Revenge and justice are far from the same thing, and most of the acts of vengeance in Ovid's poem take place on the spur of the moment and are completely unfair. The most unfair type of revenge is also the most common. This usually happens when Jupiter rapes a mortal woman; then, because his wife Juno is angry about what's happened, she typically imposes horrible punishments on the victims of the rape, instead of confronting her husband. Even when the targets of revenge seem justified (say, Tereus) there is often collateral damage (say, Tereus's son). All in all, revenge in The Metamorphoses is a messy business.
In The Metamorphoses, revenge is typically the reaction of people who are otherwise powerless.
Most of the characters who seek vengeance in The Metamorphoses are women. This reflects the fact that women in Ovid's world are usually victims to begin with.