I have been learning Japanese for about a year, and this is a thorny topic. The best is the one best for your skills and learning style. To be honest, I think you could use free resources on the internet. I also think it depends whether you want to learn formally or more organically (the latter like me).
Genki 1 and 2 are frequently mentioned and will cover everything you mention to a lower intermediate level. I find them long-winded. Tai Kim online covers grammar and it's free - I find it heavy going, but there is an application, which is lighter.
Learn the kana. You will have to tolerate reading like a 5-year old for a while, but persist. There are plenty of good apps; I used Dr. Moku.
I use Wankani for learning kanji, though it uses SRS and can be overwhelming a times. You could also try Heisig's 'Remembering the Kanji' backed up by Kanji Kohi (?) also free. If you want to write kanji, then you have a lot of practice ahead - I was advised by a senior manager working in Japan that it is unnecessary, given modern keyboarding - but they are beautiful things and you will miss the tactile experience.
My own advice is to speak the language as a priority, and I used Pimsleur and Michel Thomas, but Assimil is also recommended. After just three months of using the first two, I spent one month in Japan without problem - but then I only knew kana, so I was functionally illiterate. Speak the language (iTalki!) as much as you can. At first, I was scared of making mistakes, but now I realise that is all part of learning. In my view, this is one of the shortcomings of the Japanese method of teaching. Japanese students spend six years learning English - mostly grammar based, but in my experience very few speak it well, if at all.
I am sure you will get lots of other advice - you can spend a fortune on textbooks, and many have useful advice, but there is nothing that will beat getting the basics and speaking as much as possible!
I will definitely recommend you the Genki series. They have Genki I and Genki II. They also publish workbooks ( I & II ), too. The structure of the books is very simple and well-organized. Normally, some textbooks from famous publishers have too many letters or descriptions and they are quite confusing and hard to figure out which grammatical structures or vocabs are important because there are too much information.
I hope you will enjoy learning Japanese!
My favorite book to learn Kanji is "Kanji in context".
For each Kanji, there are example words.
This is a sample of this book: "http://bookclub.japantimes.co.jp/pdf/1529.pdf".
I like it because example words are really useful and the method is cool (first pass you learn just most important words for each Kanji and after you learn more difficult one).
Moreover there are a lot of words to improve the vocabulary. There are two workbooks but you can't use it before to learn the first 350 Kanji. It's the only drawback of this book.
Heising's Remembering the Kanji is certainly the worst kanji book. A lot of people said that his method with mnemonic is really cool.
Actually everyone should use mnemonic to learn Kanji. But it's far better to use your own mnemonics and you can invented them without a book.
Moreover it's far better to learn everything in the same time to maximize neuronal connections (sounds, strokes, meanings, compounds).
In Heising in the first book you learn the meaning, in the second you learn just one word for each Kanji with the sound.
For grammar, I don't like Genki at all, it's too scholar for me. I prefer "more technical' book. I use a french book so I can't give you any advices.
I'm more than willing to help you with the basics (I'm not sure if I can help you beyond that) if your interested send me a message. As for books if you take lessons you can sometimes request if the teacher has pdf of the book that you would like to work from. Personally I used a nintendo game called My Japanese Coach that was very fun and comprehensive to get a solid basis for Japanese it covers everything you find in the common/popular books. Such as katakana/hiragana, sentence structure, grammar, culture, Kanji, writing, listening and speaking through games that are very fun and educative, I've found it to be incredible useful and if you're serious about learning Japanese I think you should give it a try.
I agree that like most say that genki can be quit useful I have genki 1 and workbook 1and 2. (although I haven't worked in them.) I myself worked through Japanese For Busy People romanised version and I did learn a bunch of stuff and so I think it can be useful to learn some basic comprehension of how the language works. Books that should be very readible accesseble (forgive my poor typing also a second language and it's late at the moment.) are the www.mangaunversity.com series where you learn from reading manga's there's kana de manga and several (I believe 5 or 6) kanji de manga. In the series you learn kana or kanji through reading small manga like stories that are usually easy to understand and fun to learn from. I think it might also be worth it to give www.japanesepod101.com and Japaneseclass.jp a try. Japanesepod101 is a site with many podcasts (with explanations available in pdf's) for you to listen and they talk about things such as grammar, sentence structure, vocabulary, proverbs, culture and a lot of other very useful things. I have a bunch more to say but otherwise my answer is getting too long. If your interested in more in either Japanese or Dutch (I saw in your profile that you want to learn that too) send me a message.