What happens to the spinning wheels of the landing gear after takeoff, before retraction?
In most common passenger aircraft brakes are automatically applied when the gear is retracted. The wheels stop spinning before they enter the wheel well. This is particularly important because a spinning damaged tire may cause damage to hydraulic and fuel lines which are usually routed near the wheel bay. 

The nose wheels have no brakes, so they are spinning while they are being retracted. Most aircraft have rubber brake pads (snubbers) fitted in the nose gear bay that stop the spinning of the gear once the wheels are retracted.