Liquid plumber and draino mostly just dissolve soap scum. Generally they are fairly poor at attacking hair clogs (probably the main issue in a bathroom sink or tub)
1: open up the p-trap under your sink, should just be couple of nuts and washers holding it on. It's probably full of crap. You can either take it outside, and shake it out/blow it out with a hose, or dig the gross clog out with a stick.
2: if the trap is clear, and the drain is still slow/clogged, get a pipe snake and snake your wall drain when the p-trap is removed. You can get a cheap pipe snake for like $10 at Home Depot.
If you absolutely must not get your hands dirty:
Go to Home Depot, and look for 10 minute hair clog remover. Barring that, buy the stuff that comes in a bottle that comes inside a clear PVC bag. That stuff is so potent that eventually it eats through the bottle, which is why the bag is there. If the bottle bursts, we know the bag will hold out long enough for someone to notice and dispose of the product. Follow the directions, and don't get any on your skin or in your eyes. Pray that you have a hair, soap, and grease clog, and that its not a root intrusion or anything else that chemicals won't fix. Cause then what happens is that you have a drain full of caustic lye that you are going to have to roto-root through with a drain snake. Which means you'll probably get it on you.