Learning roadmap for additive combinatorics

It is difficult to say if you will be able to do this as I do not know you. But as a general advice given the way you describe your background I think to read this book by Tao and Vu in detail will be a bit challenging.

As one alternative suggestion there are two books (from the mid 1990s) by Nathanson called Additive Number Theory (Graduate Texts in Mathematics, Springer).

Now, this is called Additive Number Theory and not everything in those two books is what would go today as Additive Combinatorics, but a sizeable part is (in particular in the second book, subtitle 'Inverse problems and the geometry of sumsets'). Note that the word 'Additive Combinatorics' is a lot more recent than some of the classical results in the subject.

These being graduate level books you might still have difficulty to understand eevrything in detail, but the overall style is more accessible, while the book(s) still contain a lot of interesting material.