Grammar, logic, and rhetoric have always been considered the foundations of a classical education. The three subjects together were denoted by the word trivium during the Middle Ages (although the tradition of first learning those three subjects was established in ancient Greece). The trivium is aimed at giving you a better command over language. You start with grammar, which is used in logic, which is used in rhetoric.
Regardless of whether you're a native speaker, learning grammar is essential to improving reading comprehension and clarity of writing. Any good grammar book will get you up to speed with the basics. One particularly well-organized book is Nelson and Greenbaum's 1 An Introduction to English Grammar.
For logic, Peter Kreeft's 2 Socratic Logic is a fantastic resource for "old logic" (that is, not symbolic or mathematical logic). It has been designed for self-learners, and comes with a plethora of exercises.
For rhetoric, 3 Farnsworth's Classical English Style is a great choice for anyone wanting to speak or write with a clear, persuasive, enjoyable, and unforgettable style. It illustrates the principles of style through lots of examples; the King James Bible, Shakespeare, Abraham Lincoln, and Winston Churchill are just a few of the examples quoted.
Finally, 4 Sister Miriam Joseph's book on The Trivium is a wonderful compilation of the above three topics into a single book. 5 Wooden Books' Trivium is also written along the same lines. Personally, I would recommend purchasing the individual books, and use these compilations as a reference if necessary.