Stable atoms are eternal, but that doesn't mean that we can't destroy or create them. Atoms are immortal in the sense that, if left on their own, they will last forever (this is for the stable ones). But, given enough force, we can break atoms apart. The way we do that is usually by smashing them in the wall (figuratively speaking), and we will talk more about that later on.
We can also create new atoms (new elements on the periodic table) by smashing protons and neutrons to existing atoms and hoping that they stick. Here is an interesting article about the creation of eka radon 118. Many of the heaviest atoms in the periodic table were created in the lab.
Many of the elements with atomic numbers above 100 are synthetic (and most of them have no use). See the WebElements website to learn more about each elements if you are curious. Plutonium is a synthesized element. All the natural plutonium there was on Earth originally has decayed by now (plutonium is very radioactive). So, if you want plutonium (and - really - you don't), you have to create it in the lab which is unfortunately what nuclear reactors do. In nuclear reactors, plutonium is an unwanted radioactive waste.
Here is a Phet simulation to learn with. The goal is to build atoms by adding protons/neutrons and electrons. See if you can create a heavy atom. Look to see the difference between an isotope, an ion, or a stable neutral atom. The applet will tell you whether you have an ion or a neutral atom. You can see the symbol, the mass number, and the net charge on the right. Note that the nucleus will be jiggling if you have constructed an unstable isotope. For now, choose the orbits model of the atoms. We will learn the cloud model later on (the correct model). Make sure to try the game as well!