A stray cat came to my house and I don't know what to do..

I don't know if this is particularly controversial but last night a cat kept lurking around my backyard. Tonight when I got home, it was on my back deck looking through my screen door. I opened the door and let it in the kitchen. It's extremely friendly and behaves like a house cat. It's obviously somebody's pet. I'm not sure how I should try to find the owner.. Should I ask my Facebook friends? Take it to an animal shelter? I'm planning on taking it to the vet tomorrow to see if it's microchipped. Anyone else have any ideas of how I can find it's rightful owner? It's such a sweet cat, I'm sure somebody is missing it! 😿