Details are complex and partly off-topic for this site but understanding the logic behind EU law might help. Basically, you don't have any direct rights, only your mother does. That said, a person's right of free movement in the EU also entails e.g. traveling or moving to another EU country with their non-EU spouse or dependents.
But the idea is that precluding EU citizens to live with their children or spouse or care for their parents would hurt the EU citizens' right to move freely, not that the children, parents or spouse have any rights by themselves. Consequently, you will not generally gain any right to travel or work somewhere in the EU without your mother. In any case, you might still need a visa or residence permit where applicable, it's only that obtaining one should become easier.
Beside the useful links provided by Michael Borgwardt and Alp, the EU also has a webpage on travel for members of an EU citizen's family.