A fundamental fact about mixed strategy equilibria is that a player only mixes between two (or more) strategies if she is indifferent between the two (or more) strategies. So if you know which strategies are played with positive probability (the support), you can use this condition to constrain the behavior of the other player. This can be quite complicated. But thinking of matching pennies, it is clear that the opponent has to mix with a probability of $0.5$ and $0.5$ to make a player indifferent. In two player games, this gives you linear equations. For games with more players, you get polynomial equation and things can get very difficult.
Also, if there is a profitable deviation, there is a pure strategy profitable deviation. In principle, you can find all Nash equilibria by checking these conditions for every profitable support and then test the equilibria against pure strategy deviations.
For games like matching pennies or rock-paper-scissors, one can also make ones live easier by looking at symmetric equilibria. Nash has shown in his thesis that every (finite) symmetric game has a symmetric equilibrium, but making precise what symetric means takes some group theory.
Lastly, equilibria may not be unique in zero-sum games. An easy example is given by a game in which a player has a redundant strategy, a strategy that gives exactly the same payoff as another strategy, no matter what the other player does. If such a strategy is played in equilibrium, one can mix between the redundant strategies without changing the payoffs. But even if there is more than one equilibrium in a zero-sum game, all equilibria will have the same expected payoff. This is a consequence of von Neumann's minmax theorem.