First, a lesson in what not to use. One of the more caustic products you may have in your home is a commercial drain cleaner designed to unclog sinks. If you've ever had a clogged sink, you've probably used one of these incredibly corrosive cleaners. The typical acidic drain cleaner essentially burns the gunk in your pipes, clearing it away. Some drain cleaners are now designed to use enzyme action, which may sound better than acid, but enzymes can gradually eat away at your pipes. Injecting air or carbon dioxide into the drain is less harmful, so you might look for those products instead. Another approach is to use high-pressure water treatments. Still, quite often pouring a lot of boiling water down the drain will do the trick. Try that process first before moving on to the more advanced methods.
Once you've cleared a nasty clog, commit yourself to practicing preventative maintenance in your kitchen sink. For starters, make sure that nothing solid goes down the drain, especially if you do not have a garbage disposal. Keep a small compost bin or bucket next to the faucet to remind you to scrape those plates clean before putting them in the sink or dishwasher. Once a week, dump 1/4 cup baking soda down each of your kitchen drains, followed by 1/2 cup vinegar. Let this mixture sit in the drain for 20 minutes or more without any other liquids going down. While this is sitting, boil some water on the stove. Finally, flush the drains with the boiling water.
You might need to give the process some added oomph if a lot of greasy plates have been rinsed recently. If that's the case, mix 1/2 cup salt and 1/2 cup baking soda together, pour down the drain, and then flush with hot tap water.