If small amounts of oil that hardens when at room temp, like bacon fat. Throw it in a tin can to cool, garbage when the can fills. Oil that doesn’t harden, personally I put a bunch of dish soap into the oily pan to absorb the oil and wash it down the sink. Not sure if the dish soap does enough but seems safe to me.
If its a large amount, like for deep frying. Local recycling might take it. I know curbside pickup will take used motor oil for me, so I imagine they’d take fryer oil too.