We do share a common ancestor with plants! Everything shares an ancestor with everything else. We call it LUCA, for Last Universal Common Ancestor (sometimes called just LUA). At least, that’s what the most widely-supported theories say.
That LUCA would have resembled a very basic modern bacterium, with a circular genome (as opposed to ours, which is in 46 linear pieces) and would have lived on Earth about 3.5 billion years ago.
Eventually, some single-celled organisms gulped up small bacteria and used them as internal energy factories. Eventually, those enslaved power plants became mitochondria, which retain their bacterial-type circular genomes today. The parent cells walled off their own DNA inside a nucleus at about the same time, transforming into eukaryotes. Some of those eukaryotes then swallowed up photosynthetic cyanobacteria to go along with their mitochondria, and that was the origin of plants.
Here’s a nice little diagram of how we think it all went down, via Wikipedia:
As for the transformation from single cells to multicellular splendor? That’s a (mostly mysterious) story for a different day.