Final answer:
After starting a vehicle, one should check the Temperature gauge and Oil pressure gauge to ensure the engine is operating properly.
Step-by-step explanation:
Immediately after starting the vehicle's engine, the gauges that should be checked are the Temperature gauge and the Oil pressure gauge. These gauges are crucial for ensuring that the engine is functioning properly and not overheating or running without sufficient oil pressure, which could lead to engine damage. The temperature gauge indicates the engine's coolant temperature and the oil pressure gauge monitors the pressure of the oil circulating through the engine.