I’m not promoting any of these shopping links below. I provide them only to show you the books I’m using in my quest to learn Japanese. While Amazon was usually the cheapest option when I bought these books, I did find other sources for one or two of them. Shop around for the best price!

The Genki books are not cheap, but they have been around for a long time and are used by many schools. After starting with other books, these have become my daily learning resource. You could get away with just the main Genki I textbook, but I find having the Workbook and Answer Key adds value. Be aware..the Answer Key is written completely in Hiragana, Katakana and Kanji ..plus it ships from Japan and took weeks to arrive.
Genki I – An Integrated Course in Elementary Japanese Language

Genki I Workbook

Genki Answer Key

This is the book I started with. I found some of the exercises a bit repetitive however, and have decided to focus on the Genki textbook instead.
Japanese for Busy People – Kana Version

This is from the same authors as above and I did make use of this while learning Hiragana and Katakana.
Japanese For Busy People – Kana Workbook

This is a set of four books from beginner through to advanced. One is aimed at reading and writing, and while I did use it when learning Hiragana and Katakana, it’s pretty basic. The same goes for the three main books. I use it now primarily as a portable tool. The books are small, light and easy to pack with you and have simple exercises at the end of each lesson. Handy for practicing when you have a few minutes while out and about.
Living Language Japanese – Complete Set

Specialised books …

This book was difficult to find but was highly recommended on a number of websites. It has and continues to be, a valuable tool for learning particles. It’s well laid out and easy to use.
Japanese Particle Workbook

The other specialised books below, I’ve rarely used. Hopefully I’ll use them more as I progress beyond the beginner level.
Dictionary of Basic Japanese Grammar

Oxford Beginner’s Japanese Dictionary

Japanese Sentence Patterns for Effective Communication

The Handbook of Japanese Verbs

Kanji Books.

I haven’t started these books as of yet. I’ve learned a few basic Kanji using online resources, but after learning Hiragana and Katakana, decided to focus on  grammar and vocabulary before ‘tackling’ Kanji in a serious way.

The Kodansha Kanji Learner’s Course: A Step-by-Step Guide to Mastering 2300 Characters

The Kanji Learner’s Course Green Book: Writing Practice Workbook for The Kodansha Kanji Learner’s Course

The Kodansha Kanji Learner’s Dictionary: Revised and Expanded