Forget about pushing friction and slowing friction. Think of static friction and kinetic friction.
Static friction is friction between two or more solid objects that are not moving relative to each other. It's what keeps the car from slipping. When the car is in motion, ideally, the tyre and road surface do not move with respect to one another, the tyre grabs the road. It works the same way with the soles of your shoes and ground.
Work done is equal to Force times Distance. Since the tyre and road are not moving with respect to each other, no work is done against static friction, nor can it ever. When you are cruising at a constant speed on a level road, the engine is working against friction, but this is kinetic friction: the friction between internal parts of the car's engine and drivetrain and the friction between the car's body and the air.
There is also loss within the tyre as it rotates. The tyre flexes as different sections of the tyre come into contact with the road during rotation. The deformation is not perfectly elastic and some of the energy is lost as heat during the process. Underinflated tyres can add to the effect and increase fuel consumption. Recommended tyre pressure is a trade off between comfort and handling.
When you apply the brakes, they are designed to cause kinetic friction between the brake components (pads and rotors for disk brakes, shoes and drums for conventional) which converts the kinetic energy of the car into heat. Electrical cars can convert some of the kinetic energy back to electrical energy which is a more efficient use of the kinetic energy.
When the tyres slide, as when you go into a skid, kinetic friction between the tyre and the road does slow the car down, but nowhere nearly as efficiently as the brakes would, which is why modern cars have anti-lock brakes. Besides, steering is nil when in a skid.