Just wondering (especially in the rain) if an airplanes rear wheels will spin like a cars rear tyres will spin when it accelerates from a halt.
No, the tires on a plane aren't driven by the engine. All the power comes from the propeller/turbine pushing on the air.
Well, actually, there have been a (very) few experimental or one-off planes with driven wheels. But even then, the vast majority of them had wheels that were driven by low-power electric motors without enough torque to actually "burn rubber". Those motors wouldn't have been used for takeoff, just as a more fuel-efficient way to taxi around the airport.
If the brakes are dragging, it is entirely possible for it to skid "the other way": i.e. with the engine pulling the airframe forward while the tire doesn't rotate. But that's about it.
The airplanes obtain their forward momentum from a force called thrust, in most planes it comes from either a propeller that have airflow deflection or a turbine engines that blows air due to ignition.
The only time when the gear is rolling is after takeoff and it's stopped by pressing the brakes.