Today we learnt at school that atoms can be destructed. I believe Physics is a great science to do experiment and I would like to try it at home. Could you tell me what I need to do it? and is it dangerous?
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An atom is composed of a nucleus surrounded by small particle called electrons. Electrons are relatively easy to remove (for example to make electrical current). So people talk of "destroying" (not really the proper word) an atom when somehow they "destroy" the nucleus.
Destroying atoms is not a game or something to be experiment with for fun. It is not even the fact that it could be dangerous (it can be extremely dangerous), it is simply very difficult to control without very complex equipment.
Some atoms will transform by themselves into other atoms, spontaneously, as explained by some of the comments. As they do that they will produce particles that are often (not only) the nuclei of very small atoms (the smallest is Hydrogene, and next one is Helium). These are what is called radioactivity, and (worth repeating) it can be extremely dangerous. Protection against radioactivity can be very difficult. Fortunately, natural radioactivity from spontaneous tranformation of atoms is usually very weak (not in Fukushima, though).
Actually you do not destroy atoms, but only transform them into other atoms, either by fusion of two nuclei into a larger one, or by breaking a large nucleus into smaller ones, which is called fission.
This can produce large amounts of energy, as in atomic bomb and in nuclear powerplants.
Here you have plenty of words to search the Internet. If you want as list of know atoms (up to some details) look at Mendeleev Periodic Table. There are only about 120 different ones (up to differences that are a bit to delicate to explain here). These are the basic chemicals existing in nature.
Now you can assemble them into larger things called molecules, that constitute all material things around you. Working with molecule is chemistry. There are kit to do simple tranformations of molecules that are not dangerous (it can be dangerous too, if you are careless).
Good luck in school. That is a good place to learn. And books too.