Atoms can be broken down in several ways. One is radiation bombardment; when an atom is hit with high energy particles, it can become unstable. Unstable atoms will emit radiation in the form of an alpha, beta, and gamma radiation. Alpha radiation is when the atom emits a helium-4 atom at high velocity. Alpha radiation takes two protons in the process, however, and can reduce an atoms atomic number. For example, if a carbon atom underwent alpha radiation, it would become beryllium. Beta radiation causes an atom to convert a neutron into a proton and an electron or a proton and a positron. The electron (or positron) is ejected, and a neutrino is released. Beta radiation causes the atomic number to go up by one. Going back to the last example, if a carbon atom underwent beta radiation, it would become nitrogen. Gamma radiation does not affect the nucleus, and is simply the release of a photon.
In addition, atoms can be split through nuclear fission. For example, when U-235 is hit with a low velocity neutron, it breaks into krypton and barium, releasing three free neutrons. Atoms heavier than iron tend to undergo radioactive fission.
Radioactive fusion occurs inside the sun, and is when small atoms are combined together to form larger atoms. This process releases vast energy, and can only be carried out on earth using atomic weapons.
Electron levels, or shells as they are more commonly called, are in constant states of change. When atoms absorb energy, they usually do so by changing the energy state of an electron. Photons can be released by atoms when electrons drop down energy levels. Electric current causes atoms to pass around their valence (top shell) electrons. Atoms with more or less electrons than protons are called ions, and are very common in the world around you.
Here are some links you can use to read up on radioactivity, fusion, fission, and electron shells.
This is a page about fission reactions, showing how atoms can be broken apart.
This is about the proton-proton chain reactions that occur in stars. It details how larger elements are formed.