Since you are in your third year in Mechancial Engineering, i would assume that stuff like Linear Algebra and Differential Equations are known to you, at least the basics of it. If you feel you need revision for either of those., MIT OCW should take care of it. I would say that LA is more used in Robotics so give it more importance than DiffEq.
Since you want to know more about Planning and Control, I would really suggest the specialization on Coursera called Modern Robotics. I did this course personally and would suggest you do the same. It focuses more on the robotic manipulation side of things (robotic arms and mobile robots), but the concepts are transferable to any robotic system you want. This course is also based on the Modern Robotics textbook (that you mentioned) and uses Python/MATLAB for the assignments. Apply for financial aid for all 6 of those courses and you will have a strong background to read more advanced textbooks and research papers.
If you want to dive deep into Control Systems then, Control Bootcamp by Steve Brunton is the best place to start learning it (In fact since you are a MechE also check out his other playlists. His channel is a goldmine).
As for planning even though most of the introduction concepts are covered in Modern Robotics, you can look up Computational Motion Planning on Coursera for additional info (although iirc the only new concept in CMP is Artificial Potential Fields.)
While I understand this post was for the pre-requisites, what I feel is there is no such strong background knowledge required. Textbooks are usually not written in a friendly manner many times and a certain topic may feel difficult cause of it. If you encounter some terms you don't know, Google it to understand, and get the feel for it.
Hope this helps. Do ask if you need any more help.
Source: Final Year MechE considering a PhD in Robotics.