Don't know what model Moen you have. Check the install instructions that came with it. The common Moens, Chateau-style and Positemp, have a U-shaped bronze clip which retains the cartridge. Pull out the clip, then the cartridge simply pulls straight out.
When the job is all finished, check for "hot on the left, cold on the right". When you remove the cartridge, there is on indication which way it should go back . This is by design. The instructions tell you how to easily reverse hot/cold if necessary. ( It is done easily with the cartridge, doe not involve the pipes at all)
