Call the court where your case is scheduled to be heard to inquire what their procedures are for requesting a continuance. Most courts require at least 10 days' notice and a formal request in writing, though some may allow you to make an oral request.
Contact the opposing party to ask them to agree to the continuance. This isn't always necessary, but it is proper etiquette. If the opposing side agrees to the continuance, the process is often much easier, and your request is likely to be granted. If they don't agree, you can still make the request to the court.
File a written motion with the court at least 10 days before your scheduled hearing. Make sure to note in the motion whether you have contacted the opposing party and whether they have agreed to your request. If the court doesn't require a written motion, you can get a new date from the court clerk orally. It's typically a good idea to file a written motion in any case so there is a record of your request.
Call back the next day to see if the judge has approved the motion. If your motion has been approved, get the new court date and pick up any necessary paperwork from the court.
Inform the opposing side of the new court date if the court hasn't done so already. Typically this is done by having the new court paperwork served on the opposing side, although in smaller cases an oral or mailed notification may suffice.