Just as the phrase "raining cats and dogs" doesn't have much to do with the four-legged creatures, neither does the Spanish phrase levantar la liebre have much to do with hares - it has to do with figuratively exposing or shedding light on something. It seems that whatever the language, we like to talk about animals even when we're really talking about something else.
Here are more than a dozen Spanish phrases, most of them colloquial, that include the names of animals. You can communicate more like a native speaker if you use these phrases - just don't understand or translate them too literally!
Someone or something trying to do or be two different things at once can be said to be a caballo entre (like a horse between) those things. Turquía está a caballo entre dos mundos: geográficamente se ubica entre Europa y Asia, y culturalmente se encuentra desgarrada entre el islam y el Occidente. (Turkey has its feet planted in two worlds: Geographically it is located between Europe and Asia, and culturally it is torn between Islam and the West.)
Someone who is crazy, strange or weird can be said to be como una cabra (like a goat). Seguro que pensaron que estaba como una cabra. (I'm sure they thought I was loony.)
Como un elefante en una cacharrería (like an elephant in a pottery shop) is the equivalent of "like a bull in a china shop." No entres como un elefante en una cacharrería. Tómate tu tiempo e intenta recabar la información necesaria para conocer la empresa. (Don't start out like a bull in a china shop. Take your time and try to gather the information needed to understand the business.)
Someone who is extremely lucky by avoiding or recovering quickly from disasters can be said to tener más vidas que un gato (have more lives than a cat). El joven ciclista demostró que posee más vidas que un gato. (The young bicyclist showed he may get knocked down but is never out.) Incidentally, while we often talk about cats having nine lives, Spanish speakers seem to think they have seven or nine.
If there's a hidden or secret reason for something occurring, we might say aquí hay gato encerrado (here there is an enclosed cat). Sometimes the phrase is the equivalent of "there's something fishy going on." The phrase may have come from centuries ago when money was sometimes hidden in a small bag made of cat's fur. Supongo que Pablo se daba cuenta de que aquí había gato encerrado, pero no sabía nada de nuestro secreto. (I suppose that Pablo noticed that something unusual was happening, but he didn't know anything about our secret.)
To do something daring or risky - often when nobody else is willing - is to ponerle el cascabel al gato (put the bell on the cat). Similar expressions in English include "to take the plunge" or "to stick one's neck out." This phrase is quite common in political contexts. Después de seis años de dudas, indecisiones, explicaciones y excusas, el presidente finalmente le puso el cascabel al gato. (After six years of hesitation, indecision, explanations, and excuses, the president finally took the plunge.)
Hares were once far more valuable than cats, so dar gato por liebre or meter gato por liebre (providing a cat instead of a hare) came to mean to swindle or dupe someone. Me dieron gato por liebre cuando intenté comprar mi móvil por internet. (They ripped me off when I tried to buy my cellphone online.)
To lift the hare, levantar la liebre, is to reveal a secret or something that had not been known. In English we might let the cat out of the bag. Era la atleta que levantó la liebre del dopaje. (She was the athlete who who exposed the secretive practice of doping.)
If someone can see extremely well or is very good at noticing fine details, you can say that person has the vista de lince (lynx's eyesight) or ojo de lince (lynx's eye). It's just as we can talk about someone being or having an eagle eye. The word for eagle, águila, works in these phrases as well. Uno de los voluntarios, que tenía un ojo de lince, descubrió el abrigo de la niña en el bosque. (One of the volunteers, who had eagle eyes, found the girl's raincoat in the forest.)
If you believe someone is lying to you - or, colloquially, pulling your leg - you can respond with a otro perro con ese hueso (to another dog with that bone). ¿Me dices que estudiaste toda la noche? ¡A otro perro con ese hueso! (You're telling me you studied all night? Baloney!)
In English, you might sweat like a hog, but in Spanish it's sweating like a chicken, sudar como un pollo. Esa noche sudé como un pollo. Creo que perdí dos kilos. (That night I sweated like a pig. I think I lost 2 kilograms.) In Colombia, a popular sauce-covered chicken dish is known as pollo sudado (sweated chicken).
In English, if we're slow we might do something at a snail's pace, but in Spanish it's a turtle's pace, a paso de tortuga. Los trabajos para la construcción del nuevo mercado público marchan a paso de tortuga. (Work toward the construction of the new public market is proceeding at a snail's pace.)
If something is more of the same to the point where it becomes irrelevant or nearly so, you can call it one more stripe for the tiger, una raya más al tigre or una mancha más al tigre. Aunque para muchos es simplemente una raya más al tigre, me importa mucho su compromiso. (Although for many it doesn't make much difference, her promise matters a lot to me.)