All cells receive the same DNA sequence during early development of the organism. Knowing this, why do some cells become bone cells and others become muscle?

A process called cellular differentiation occurs. Although every cell contains the same genetic material it doesn't express every single gene.

For example muscle cells don't secrete adrenaline even though they have the genetic code for it. There are many processes that revolve around gene activation and inactivation that are studied by a branch of genetics called "epigenetics". Some examples of processes that inactivate genes are methylations or acylations which represent the attachement of different functional groups to the nitrogenous bases.