Current estimates of the number of stars in our galaxy (the Milky Way) are between 100 and 400 billion (being inside the galaxy makes it difficult to count them all, a bit like being in one spot in a forest and trying to count the total number of trees). Assuming that our solar system has an average number of planets (8), that makes about 800 billion to 3.2 trillion planets of various sizes. Of these, the number of Earth-sized planets is estimated to be in the 10s of billions.
So far, around 4 thousand extra-solar planets (or “exoplanets”) have been detected. There are still plenty more left for a budding astronomer to discover and name. The next big challenge will be to detect if any of them are capable of supporting life!