For a single slow sink or basin, put a bowl under the trap, unscrew both ends, empty the contents into a bin bag (never back down the drain) and scrub it out. This alone fixes the majority of household blockages. For baths and showers, the cause is nearly always a hair plug — fish it out with pliers or a Zip-It tool.
For an external gully, lift the grid, scoop the debris out by hand with gloves, and flush with a hose. If the water level does not drop, the blockage is downstream and needs rodding or jetting.
Stop and call a professional if water is backing up in more than one fixture, if sewage is escaping outside, if a manhole is full, or if you cannot identify where the blockage is. Pushing on at that point risks flooding the property.