Within limits, it really is true if you do not take care of yourself, no one else is going to. This old adage fairly well defines parenting tasks related to teaching children to fight and fend for themselves. First, your child needs to learn to assert and protect her rights. Next, she must learn there are limits to assertiveness. Finally, she must learn when to trust other people to not harm her and when to turn her safekeeping over to others. In the bicycle riding example, your preschooler or grade schooler needs to assert her right to try to ride the bicycle alone. Next, she needs to accept the reality wrecking the bicycle or hurting herself are not worth making the point she can do it herself. Finally, she must accept a little assistance in steadying herself.
Your preschooler should definitely assert her right not to have her toys taken away by a younger brother. She must learn, though, not to shove little brother or hit him in the head. Finally, she must accept the reality these types of disputes are sometimes best mediated by Mom or Dad. You want to be very sure, though, she does first try asserting and protecting her rights on her own. If she is excessively aggressive, respond only to the excesses and not to her asserting and protecting her rights.
If two children get into an argument and immediately run to you to tattle on each other, tell them they have to work on the problem a while longer before you are willing to intervene. (It is an interesting and frequently overlooked fact, when two children are in an argument, it is usually the younger child who started it. Typically, the older child gets blamed, when the younger child has actually caused the problem to begin with.)