I'm on birth control and my period has been starting early. When that happens, should I continue the pills like normal or start taking the sugar pill?

Continue. You should continue your pills as prescibed. If your periods start early every cycle, you may need a different birth control pill.
Take as prescribed. It is common to have some changes in your bleeding pattern on birth control pills. Sometimes it is in between periods (breakthrough bleeding), and sometimes you can bleed more or less often. Do not start the sugar pills until it is time as that will only make the bleeding more irregular and more important will decrease their ability to prevent pregnancy.