If the trap doesn't have a pull-out cleanout, you might have to wrench free the P-trap or use a sink plunger to clear the clogs. You also want to check for leaks and replace the faulty components.
Watch this video for chemical-free ways to fix a clogged drain ... it can be something stuck in the p-trap. The drain, located under your sink is named for the shape it makes if you look at ...