On this FAQ page of our guide to Animal Crossing New Horizons you will find out where to look for fossils.
Every day there are four places on your island where you can dig out some fossils. How can I find fossils? Look for large stars on the ground, the ones shown in the pictures above.
After you dig this place up with a shovel, your character will get a fossil after a while. You can donate it to the museum, store it (or display it) at home or sell it. We recommend the first two solutions.