I think chaliye is used to show respect...

chalo shuru karein chalo shuru karte hain if u r saying it to many people say "chaliye" instead of chalo

as tushar said, you can say it in both ways -- चलो/चलिए, शुरू करते हैं! चलो/ चलिए, शुरू करें! चलिए is generally used for showing respect to other person or in a formal situation. you can also use आरंभ instead of शुरू, e.g. चलो/चलिए आरंभ करते हैं (or करें)!