New, cheap mountain bike: What's most likely to break?
Information about the specific components would help. I am not sure what will break first, but the derailleurs will probably be the first component to need servicing - especially if you purchased this bike from a sporting goods retailer instead of a bike shop. Brakes will probably be needing some love in the near future. If you ride on trails at all the fork will probably end up needing some maintenance quickly. The wheels will also probably go out of true pretty quickly as well.  

One thing you can do to help keep the bike in good running order is to frequently put chain lube on your chain to keep it, your derailleurs, and cassette well lubricated.