kapil1089thekin, since you are experienced in C, why do you think new will be any different from malloc? What you new you must delete.
However, to help you out on the way, we have smart pointers in C++. Look up std::shared_ptr and std::weak_ptr. std::shared_ptr is the default one you should use. In (doubly) linked lists, you must watch out, because if the prev pointer is a shared_ptr and the next is a shared_ptr, you will get circular references and neither object will be destroyed (they will keep each other alive). For this situation, use weak_ptr on the prev and shared_ptr on the next.
Experiment a little. Get a feel for smart pointers. They are powerful tools and will relieve you of the freeing duty.