Add the fillets to the pan, being sure to leave space between the fillets. Work in batches if necessary. Depending on fish and pan size you may be able to cook anywhere from one to four fillets at once. The fillets should sizzle the moment they touch the pan; if they don't, remove them and wait for the pan to get hotter.
Cook the fillets until browned on one side, about 3 minutes, flip them over and cook them until they've browned on the second side and the fish is opaque and flaky in the center, about 3 more minutes. Timing here really depends on the fish, so use your judgement over the clock.
Remove the fillets from the pan and hold them on a warm baking sheet or tray. Repeat with the remaining fillets, adding additional vegetable oil between batches if necessary. Serve pan-fried fish hot with wedges of lemon or the sauce of your choice.