For the most part, this is not a very detailed fix and can be done easily by either plunging, using a commercial or homemade drain cleaner, or by using a plumbing snake.
However, if you have a garbage disposal that could be part of the problem instead of the solution. If you do have one, but still find that food remnants are clogging your drains then it may be time to take a look at the disposal.
Determine what it is working properly, and if it has been cleaned and still the issues persist, then it may be time to replace.