How do head- and tailwinds affect airspeed?
I'm wondering how to calculate the actual speed based on that.


Which actual speed? All of the speeds you mentioned are actual speeds. None of them is in any way artificial, virtual, or otherwise invented.


  Is it the groundspeed plus/minus the headwind/tailwind?


That would be airspeed, i.e. the speed at which the plane moves relative to the air around it.


  Meaning, if groundspeed is 500mph, and the headwind is 200mph, does that mean we're only traveling 300mph?


No. If groundspeed is 500mph into a 200mph headwind, this means that the plane has to fight against the headwind, i.e. its airspeed is 700mph.

Have you ever gone swimming where there is current? Can you imagine riding your bike on a treadmill?


  Meaning it will take us 10 hours to travel 3000 miles?


No. If the groundspeed is 500mph, it will take you 6 hours to travel 3000 miles over ground.