Start with the trap. Put a bowl underneath, unscrew both ends by hand, empty the contents into a bin bag and scrub it out with a bottle brush. This clears most kitchen blockages in ten minutes.
If the trap was clean, the deposit is in the horizontal run. A hand snake fed through the waste outlet will break it up; badly furred pipework needs jetting to restore the full bore.
Prevention is straightforward: scrape and wipe pans before washing, fit a sink strainer, and never pour cooking oil down the sink even with hot water running.