First, climbing is not only normal but essential for your toddler. Second, you can do things to make your home more safe, such as fastening furniture to wall as suggested by Gamma G. Keep all chairs pushed under table or if she can pull them out place them far from the table. Put your TV on top of a high object, such as the fireplace mantel, a tall book shelf that is fastened to the wall, a chest of drawers. Keep everything away from the fish tank that she can climb on to get on top. My daughter had to keep her step stool on top of the refrigerator for awhile.
Then arrange safe areas in which she can climb. When she starts to cllimb anywhere that is unsafe move her to the safe area. My daughter stopped her kids from climbing up on the counters in the kitchen by consistently stopping them and taking them into the living room where they could climb on the sofa, food stool, a low cabinet. The sofa was backed against a wall so that they could climb up onto the back of it.
Also remember, when you start to panic about injury from a fall, that their bones are still pliable and are very unlikely to break. They can injure their brain if they fall from a height onto a hard surface but brain injury actually rarely happens. If you provide places for climbing that are on a carpeted surface it is unlikely that your toddler will be seriously injured. A busted lip is par for the age.
Let her climb the wall at the zoo with you standing right behind her. Find places that she can climb and she will be more apt to obey when you stop her from climbing in other places.