Yup it's wild geranium which is a native. You can move it but note it does seed around. Personally I'd leave it in an area of the woods you can see it and let it seed around out there. Soon you might have a nice patch of it to enjoy out there.
Are you in the PNW? It's an invasive weed out here. (they're trying to irradiate it in the Colombia gorge). It has the nickname "stinky bob" because the crushed leaves give off a small like motor oil.
lol to "stinky bob" - you know i'm going to go take a wiff tonight now don't you! I live in the northeast, wonder if it's invasive here too. It really is quite pretty! I will take Foxi's advice and leave it where it is.
Stiinky Bob, aka Herb Robert, is an entirely different plant. The leaves have more of a fern-like appearance and are less substantial.
The one pictured is indeed a wild geranium, Geranium maculatum. For a test, you can tugging on a small plant. Bob/Robert will yank right out of the ground, whereas the geraniums develop a huge and solid root system compared to the size of the plant.
Moving it to your garden is your decision, but I only allow mine in my woodland and cottage garden areas. There it colonizes in large drifts and looks good when blooming. BUT, the plant isn't astounding pretty and has short blooming period compared to other stuff out there. It also reseeds and pops up all over the place. Definitely not the best choice for a formal or even tidy garden.
Fairy tales do not tell children the dragons exist. Children already know that dragons exist. Fairy tales tell children the dragons can be killed. - G. K. Chesterton