I do not know how or when all morals develop, but here is what I know about two.
As soon as you are born you are taught empathy or not to have empathy through your parents. It is important to note, with regards to environmental adaptation, that a child born into a very dangerous and violent society would have trouble surviving and reproducing if it is taught to empathize. It would get taken advantage of and be ridiculed as weak. Likewise, a child with no empathy will have trouble surviving and reproducing in an empathy-based society because it would likely end up removed in one way or another. The parents are a good evolutionary shortcut to what the child needs to survive in the current environment.
3-5 years old is when kids usually learn that someone can lie, and usually start trying it themselves.
A guideline I have learned: If they hold you to a rule, they are now bound by that rule. If you are breaking the rules, they will learn to ignore them just as you do.
Ultimately, kids will learn moral and immoral behavior from their parents, peers, and others that spend time with them. If kids are constantly lying, usually someone is constantly lying to them, or avoiding certain issues in my experience. Ignoring is, in a sense, a form of lying, in that someone knows something is happening but is pretending that they do not. In other words, ignoring the truth. I have experienced this firsthand. Lying children with parents lying to them. When I pointed out the truth I was attacked by someone in my "family". Fun.
Response to Erica's comment under Nelson's answer. Kids, particularly before the age of 5, will 100% be affected by all choices made by the parents that involve them in any way. Of course it will shape them. Parents have full responsibility for their kids' development unless there are extreme circumstances such as full brain dysfunction. If they hang out around the parents, their mind is imprinted with those interactions. If they hang out at daycare, their minds will be imprinted by those interactions. It is well known that brains absorb, and that children are 100% dependent on their parents.