A hand-crank auger clears most household clogs. It consists of a metal cable with a widened tip that grabs or pushes debris. Power-operated augers work quicker, but a fast-turning auger can get stuck so tightly you cannot extract it.
You can run an auger through the sink and into the trap rather than dismantling the trap. However before doing so on a bathroom sink, you'll need to remove the pivot. It's usually not possible to auger a kitchen sink because of the garbage disposer and because the drain assembly for a double-bowl sink has too many turns.
If a tub drains slowly, hair and soap sludge may be stuck to the drain assembly. A plunger-type assembly is less complicated and less likely to clog than a pop-up assembly, but both need occasional cleaning.
A toilet auger is the solution when plunging fails. Otherwise remove the toilet to auger out the clog.
About 2 hours to dismantle a trap and auger a branch line
Groove-joint pliers, hand-crank auger
Dismantling and reinstalling a trap
Make the worksite comfortable; place a bucket under the trap to catch spills
Replacement washers for the trap