Cinderella was not the only child to have problems with sibling rivalry, but this age old tale shows that it is not a modern issue. This is a real family matter and parents with more than one child learn quickly that it needs to be dealt with daily, even hourly, in their homes. The first instinct is to react, usually to spats or an outburst by one or another of the children.
It does help to step back and look at the situation from a more objective perspective: Our children did not choose the family they live in!
Remember that it was your choice to have another child or children, so it is your responsibility to manage the situation and not expect the children to figure out how to get along. Creating an action plan and then sticking to it will provide structure to what can seem chaotic at times.
Children want and need the attention of their parents and will do whatever works to get that attention.
Keeping in mind the child's goal (your attention), be careful how you react to behavior. Try to catch each child doing something right and reward your child with a smile, or a hug and positive comment on what she did right. If you only scold when something incorrect happens, and that is the only attention the child gets, she will do whatever gets a response from you!
The old axiom "what you permit, you promote" is so true with children.
If you allow the children to speak disrespectfully to each other at home, you will not effectively stop them in public. If you ignore the older child when he repeatedly takes toys away from the younger one, it will be a battle to stop a new incident. If bedtime for your toddler is whenever, then do not expect any easy transition to a structured bedtime when the child starts school.
Children will challenge you to treat them "equal" and "be fair."
The truth is that they are not the same age, (unless they are twins) so should have different bedtimes, and levels of responsibility. Household chores should be tailored to each child's age and skill levels. Rules should be fair, however, and consequences the same for all. Consistently experiencing the same result for breaking a rule helps the children to respect the structure you are enforcing in the household.
Teach the children by your own actions.
Children learn what they live and see. They are great mimes, and even a two year old is watching how YOU interact with others. If you are a screamer, you are teaching your children to scream at each other. If you are neat, and always put things away, it will be easier to teach them to keep their toys in the proper place.
Provide opportunities for the children to say nice things about each other.
A creative young mother has devised something she calls "The Birthday Game." On each family member's birthday, that person is the star of the day. Signs are made and each sign highlights something positive about the birthday person like "Johnny is a good helper," "Mommy is a good storyteller," or "Susie is patient."
Then before the family sings "Happy Birthday" to the birthday person, everyone takes a turn giving examples of the positive traits of their family member. This ritual provides a powerful experience for each person present. The "star" of the day is often surprised at how their actions are noticed and appreciated, and the supportive others are taught to see their family member in a positive way.
Taking sibling rivalry seriously and addressing the issues in a positive way will go a long way in promoting family harmony, which is what you and your children really want and need!