I originally started this as a basic list of references for foreign students in our lab who are motivated to study the Japanese language, though I admit it is probably impossible to follow the "serious student timeline" I give below if you are a graduate student. That said, for undergraduate students with a bit more time on their hands, or really for anyone studying the language seriously, I offer the following basic thoughts based on my own study experiences.
We begin with the assumption that the learner has no prior experience, but wants to become "fluent," or at least wants to eventually pass JLPT N1.
It seems like a lot of textbooks out there try to make the process as "fun" as possible, but this of course is not practical for serious students who want to cover a lot of information quickly.
While there are a lot of options, I recommend the following:
GENKI I: An Integrated Course in Elementary Japanese (Textbook I and Workbook I)
GENKI II: An Integrated Course in Elementary Japanese (Textbook II and Workbook II)
Work through both of these textbooks, do all the lessons in the textbooks, and all the exercises in the workbooks. There is English, so it's very easy to get into, and covers all the basics (albeit very basic, everything is fundamental).
Once you're gone with GENKI, it's time to tackle intermediate grammar textbooks, while simultaneously learning to read/write kanji and building up a large vocabulary.
While it's fine to use English for the elementary materials discussed above, from the intermediate level it's absolutely worth going into a textbook that is essentially all in Japanese. I personally used (and really enjoyed) the following:
Authentic Japanese: Progressing from Intermediate to Advanced （中級から上級への日本語）; Textbook + Workbook
The grammar covered is good, and it gives great reading comprehension practice. In parallel with explicit kanji/vocab studies, it should be doable fairly soon after you've covered the elementary material.
There are many horrible kanji textbooks on the market, in particular ones that attempt to associate silly mnemonics with each and every character. This is completely impractical, it doesn't scale to the volume of characters one needs to memorize, and of course it's not how Japanese people learn the characters, either.
The following textbook is the gold standard for learning kanji and building up strong vocab, and I feel must be used by any serious student:
Kanji in Context (Reference book + Workbooks Vol. 1, 2)
If you've learned roughly 300-500 basic kanji covered in elementary courses, then you'll be set to start with this book, which covers all the standard-use characters (roughly 2,100). I went through every character, and every associated word, and wrote them countless times each. It takes a long time to get through this book, but once you have (and have done one thorough review afterwards), you're essentially "done."
The "standard" test that Japanese learners always take, which tests basic ability to read, grammar, vocabulary, and listening, is the Japanese Language Proficiency Test (JLPT). It seems daunting when you get started, but while working through the above study materials at a good pace, if you get some textbooks for this test, and simply do a lot of (timed) practice runs, it's totally manageable.
If you're taking things at a leisurely pace, you'll have to judge what level you should try on your own, by looking at practice problems. Prior to revising the test in 2010, there used to be a massive gap between level 3 and 2, but now they've added an extra level so apparently the N3 - N2 gap isn't as large as the old 3 - 2 gap.
In any case, for serious students working at a good pace, considering the cost of the exam I would recommend skipping N5/4/3 and going straight to N2 as your first objective. To get to this point will probably require a year or more of hard-driven studying, of course. If you can pass N2 confidently, then N1 isn't too far beyond that.
I won't recommend any textbooks for this one, as they're really all about the same, and doing as many practice problems as you can is the only way to prepare explicitly for it, so I would say "do as many as time and budget constraints allow."
There is a "Business Japanese" test, and it is kind of unique and interesting in that it really just demands you to be extremely good at listening and reading business-related materials. The level of the listening is fairly high if I recall, but for anyone in engineering or the sciences, this kind of test isn't particularly useful, really just a kind of fun practice.
For people who like Kanji, the standard is Kanji Kentei, a very fun exam. There is a whole load of textbooks and workbooks prepared by the organization giving the exam, which are nice, but if you can only pick one series for preparing for this exam, make it the books from Takahashi shoten （高橋書店）. For example, the typical texts to use are 漢字検定X＊級［頻出度順］問題集, with ＊ being your appropriate level. If you're at JLPT N2 or N1 level, you should start with 準2級, then 2級. Beyond that things get pretty rough, only for the true enthusiast.