Shinzon's motives are stated repeatedly in NEM. Destroy Earth and cripple the Federation. Earth is the capital for the UFP and HQ for Starfleet. The Romulans at the beginning of NEM were appealing for a war before the senate and Preator. However as the Romulan Praetor stated "the senate has considered Shinzon's proposal and rejected it". Later Commander Suran says that he and the other collaborators of this coup supported Shinzon because he promised action. Shinzon just took his time on actually carrying out the plan.
I suggest rewatching the scene where Shinzon has Picard prisoner onboard the Scimitar. Shinzon appears to have changed his mind from being curious about Picard to now despising him. Saying that "my life is meaningless while you're still alive". There are some other good quotes from that scene and later in Picard's ready room with holo-projector Shinzon. It became a grudge match between two men. Shinzon's plan to destroy Earth was initally to instigate a war between the UFP and the RSE. It later becomes a way to wipe out the life and legacy of Jean-Luc Picard and cement Shinzon of Remus in the history books as the man who destroyed Earth.
Does the plan make complete sense? Not really. But hey what Star Trek film doesn't have it's bonehead plans by villains who we are supposed to believe are intelligent. Looking at you Khan from TWOK, Borg from FC, Ru'afo from INS, Nero from ST09 and John Harrison from STID.