I'm new to mechanics, so this may not seem like a good question. When the gas pedal is first pressed, how does a piston start the intake stroke? Does it have some direct connection like a rod?
Judging from your question, you are indeed new to mechanics. The pressing of the gas pedal does not start the engine. The engine is already running, disconnected from the wheels by a slipping torque converter (automatic transmission) or a disengaged clutch (manual transmission).
The engine is nowadays started by using an electric starter motor powered by a battery, although newer hybrid vehicles use a traction motor to also start the engine, lacking a separate starter motor. The action to operate the starter motor is pressing the engine start button or turning the key on the ignition lock.
Pressing the gas pedal allows merely more air in. In today's vehicles, the gas pedal is monitored electronically, causing the control system to open the throttle valve more (assuming gasoline engine here), letting more air into the engine. The air flow meter notices this, and the system injects more gasoline. More air plus more gasoline results in bigger bang, speeding up the engine rotation rate and propelling the car forwards.
Of course, the connection to the wheels needs to be made to actually move the car forwards. In an automatic transmission car, speeding up the engine causes significantly more torque on the wheels through the torque converter. In a manual transmission car, you stop pressing the clutch pedal slowly, causing the clutch to slowly engage.
Diesel engines lack the throttle valve, and merely control the injected diesel fuel amount. More pressure on the gas pedal equals more diesel fuel injected.