Many new students to Japanese are excited when they learn about romaji. They think “hey, this is cool! I can learn Japanese without having to read Japanese!” After all, reading “konnichi wa” is easier than reading “こんにちは.” However, it is all too often that students become dependent on the romaji translation and end up having their progress halted as a result.
Some textbooks do away with romaji after all the hiragana and katakana are introduced. Other books provide romaji translations for every exercise in the book. Unfortunately, some Japanese textbooks are only written in romaji, which results in a student who can speak a little Japanese but cannot read or write any. Of course, this wouldn’t be too bad in and of itself, but by the time the student completes the textbook, it will be difficult to find another textbook with which to continue studying because the student will be unprepared for a textbook that uses hiragana and kanji. So in this case, the only option would be to go through another beginning Japanese textbook (one that uses hiragana) and “relearn” everything.
You will never actually encounter romaji in Japan, so the only purpose it serves is to help you cheat while studying.
You should avoid any textbook or study materials that are written entirely in romaji. A textbook that completely eliminates romaji after the hiragana is taught is ideal, but may be a bit intimidating for many people. A good compromise would be to get a book that has hiragana and romaji, and then cross out the romaji with a permanent marker when you are confident in your reading abilities.
Fortunately, even most books that contain both eventually drop the romaji when you get to advanced lessons or the second and third volumes.