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You will need to have 4 jack stands. Raise the car to ensure the OIL PAN is level, this MAY mean that the car is slightly higher in the rear ( a lot of discussion around here on that!). The drain plug is by the bat wing (Under engine cradle) on the driver's side. Remove plug (a ratcheting wrench works best) and let drain... a while, at least 45 minutes. Reinstall plug. Remove oil filter, fill new filter, lube filter seal, install hand tight. Fill with Mobil 1 ( or whatever YOU decide is better) takes about 6.5 qts. Check underneath to ensure no leaks.
Place Vette on ground. Take ignition switch to ON, but do not start engine. Scroll through the "Trip" options on DIC, when you get to the Oil Life Monitor, press and hold the RESET button, the OLM should reset to 99%. You can also, pump the gas pedal 3 or 4 times while the ignition is ON and the engine is off, should do the same thing.
Go for a spin. First time, it might take you 1.5 hours, with the bulk of that time allowing for the drain. Do NOT overfill, better to add a little later than over fill.