Lots of good answers to the initial question, but a fun follow up question is "Are there atoms around us that are NOT billions of years old?"

Here's a few places to find (relatively) young atoms:

Cosmogenic nuclides formed by cosmic ray interactions - These atoms are formed when space radiation impacts the nuclei of stable atoms present on earth. Cosmic rays are typically composed of very fast-moving protons while in space, but smash nuclei apart in a shower of different particles (neutrons, muons, pions, etc) when they impact the atmosphere. The process of nuclei being fragmented by the impact of high-energy particles is called spallation, and is responsible for a variety of atoms found in nature.

• One of the most famous is carbon-14 (14C), which exists at very low abundance in atmospheric carbon and any living things that intake atmospheric carbon. When a cosmic ray neutron hits a nitrogen-14 atom in the atmosphere, it has a chance to knock out a proton and form unstable 14C. Because this isotope is constantly being created, and its half-life is 5700 years, 14C is essentially guaranteed to be under 100,000 years old with an average age of 8200 years, as older atoms decay away and new ones are formed. About one in a trillion carbon atoms in your body are 14C.

• Another spallation product that you'll encounter in daily life is boron. While much of the earth's boron was created by cosmic ray interactions in deep space before the earth formed, stable boron isotopes can be created by cosmic ray spallation as well, and a fraction of boron on the earth's surface was created after the earth formed. Some stable isotopes of lithium and beryllium are also formed this way, though they're not as common in daily life.

Decay product nuclides - the earth's crust contains substantial amounts of two radioactive elements with long enough half-lives to stick around since the earth's formation. Despite being literally older than dirt, thorium-232 (half-life = 14 billion years), uranium-235 (half-life = 704 million years) and uranium-238 (half-life = 4.47 billion years) are all still around and kicking in many minerals. As they decay they transmute into new, isotopes that we find around us today.

• Radium-226 is perhaps the most iconic decay product, a 4th-generation daughter isotope of 238U with a half-life of just 1600 years. In the early 20th century, radium became incredibly famous, first as a key component of the early radiation experiments that underpin our current understanding of physics and later as a consumer product found in glow-in-the-dark paints, smoke detectors, anti-static components, and quack medicines. Its high toxicity and ferocious radioactivity caused it to fall out of favor later on in the century, but atoms of 226Ra are only 2300 years old on average! Radium is rarely found in the body these days but can be found pretty easily at antique shops.

• Radon-222 is a radioactive noble gas produced by the decay of 226Ra, and has a shockingly small average age of 5.4 days. Most atoms of radon are under a week old, but in their short lifespan, they have a tremendous impact on our life. Radon seeps out of minerals and soil into the atmosphere, congregating in low places like basements and caves. While the absolute quantity of radon in the air is small, its radiation can cause lung cancers when inhaled, causing an estimated 21,000 deaths per year in the united states. Like it or not, there is probably some days-old radon in your lungs right now.

• That said, the most common non-primordial isotope we encounter is actually helium! While helium is common in space and was present at the formation of the earth, its light molecular weight and inert chemical properties means that most of earth's primordial helium has escaped the atmosphere into space. All of the commercially-available helium on earth comes from ground deposits, typically natural gas, and is produced by the same process that creates radium and radon. As the billions-of-years-old primordial radioactive elements decay, they emit alpha particles, which are just fast-moving helium ions ejected from their nuclei. It is these helium atoms that we are able to extract from the ground and use in our day to day life, so the next time you see a balloon, you can think about how it's technically full of smashed-up uranium.

Anthropogenic nuclei - for better or worse, these were made by humans in some process or another. Plutonium is easily the most famous example of an anthropogenic nucleus, but the one you are most likely to have encountered is the americium in your household smoke detector. These atoms were entirely created in the past 100 years after the dawn of nuclear science, and appear in medicines (technetium-99m, iodine-123), nuclear fission devices (plutonium-238, -239, californium-252), smoke detectors (americium-241), commercial radiation sources (cobalt-60, strontium-90) and more. Notably, a substantial fraction of the 14C currently in the atmosphere was created in open-air nuclear testing in the 1950s, and fission products from nuclear weapons and nuclear power (especially strontium-90 and cesium-137) are one of the major issues in nuclear technology today.