It might be rear derailleur adjustment. Try flipping the bike upside down so you can turn the pedals and rear wheel freely, then watch the rear gears/derailleur as you "pedal" with your hand to see what exactly is rubbing/making noise. If the rear derailleur pulley closest to the gears is not perfectly lined up with the gear, the chain can rub against the adjacent gears and the rear derailleur needs some adjusting.
You may be able to just adjust the "barrel adjuster" up front, right where the cable housing meets the (plastic?) shifter body. For full details, check out a site like parktool.com
It might also be that your rear derailleur and chain need to be lubed.