Before a flight, what are the most important things to check in an aircraft (walkaround)?
Many things are checked on a walkaround. The pilot-in-command should do the walkaround, not the co-pilot. The main classes of items are as follows:

structural integrity; the aircraft must be sound and undamaged and ready to fly

sensors; sensors such as antennas and gauges should be undamaged and not have anything blocking them or adhering to them

clean flight surfaces; the control surfaces and flight surfaces must be smooth and clean; for example, the wings must not have ice on them

no leaks; hydraulic fluid, oil, fuel, and other critical liquids should not be leaking out of the aircraft

chocks, control locks and covers removed; all chocks, protective coverings and locks should be removed before flight and stowed

correct tire inflation; wear on the tires should not be excessive and the tires should be inflated properly

fasteners present and correct; all bolts, shims, nuts and other fasteners should be present, undamaged, and correctly installed

hatches and fairings secure; removable inspection plates and any other small doors, fairings and cowlings should be in place and secure.

lights ok; the lights on the aircraft should operate correctly and their lenses should have no cracks or other defects

blades ok; the propeller or turbine blades (as the case may be) should be perfect with no cracks or chips

Overlooking any of these sorts of items can cause (and has caused) serious accidents or incidents in flight or during takeoff and landing.