Can I know if an object will slip or will accelerate forward when it is pushed by a force that exceeds the maximum force of static friction?

There are static friction in many places. Both between wheel-and-road but also inside axles and such.

When a car moves without slipping/skidding over the road, then it's wheels are rolling. Rolling requires static friction! When you push your car so it rolls forward, then you have not overcome the static friction between wheel-and-road. Otherwise the wheels would not roll. Static friction is a necessity for rolling. Static friction is what grabs the bottom-part of the wheel, which is in contact with the road, and makes the wheel turn.

The only thing you have to overcome is other types of resistances such as static friction in axles and bearings but also losses due to soft tires, a soft road (try pushing the car on a sandy beach), an uneven road etc. All such resistances and losses are collected into what is called rolling friction or rolling resistance.

Ideally, there is no rolling resistance. So, ideally, it would take you very little (ideally, negligibly little) force to make the car move over a flat road. But in reality there of course always is some rolling resistance, although axle and bearing friction usually are small factors in newer cars.

When your applied force equals the maximum static friction force, the can will start to move. When that happens you are then pushing against the kinetic friction force, which is generally less than the static friction force. If you continue to apply the same force that overcame static friction, the can will accelerate since your force will be greater than the kinetic friction force. If you reduce your force to equal the kinetic friction force, the can will move with constant velocity, the velocity it had before you reduced the force to the kinetic friction force.

Hope this helps.