I tripped while carrying some laundry and tried to catch myself and ended up jamming my wrist and knuckles on a door, not sure if I should go to doctor. They are very swollen and bruised and am having strange shooting pains in my middle finger

A doctors visit is a good idea. If there's swelling, bruising, and pain, you should get it checked.

I have to wonder why you would even have to think about going to the doctor based on those sysmptoms. Yes it would be a good idea to go.