I went with Minna no Nihongo and really liked it, but I know others who had great luck with Genki, so whichever you prefer should do just fine as long as you’re able to motivate yourself to complete it.
Agree w/@fallynleaf that it’s a personal preference, and the Tofugu review is worth reading.
Main differences:
The Genki textbook has English in the textbook, but not the answers (in a separate volume), whereas Minna no Nihongo has all Japanese (vocab and grammar explanations in English are in a separate volume) while the answers to the drills are included at the back of the book in a pull-out paper book manual.
There are other differences in pedagogical approach, but as a beginning learner what is more important is that you stick to whatever resource you choose so you can make progress.
(I used both. Personally preferred MnN, but also used it after Genki to review everything and strengthen my understanding, so I knew what to look for and was ready to do it quickly.)
Is your language school in Japan? If so, it might be more likely that they use Minna no Nihongo (though if you are able to double-check, that would be best). It is generally best to use the same series for N5-N4, because MNN and Genki cover the material in a different order, so if you go from Genki 1 into MNN 2 (or vice versa), you might find yourself with some holes in your learning that you’d need to fill somehow.
I’ll add that MNN won’t really cover speaking at all. I found this to be actually a huge point in its favor for my purposes because it made it really easy to use as a self-study resource, but I’m also not interested in learning how to speak the language. The core textbook and main workbook also do not really cover kanji at all, which I also found to be a point in its favor because I was using WK to learn kanji instead (there is a MNN kanji workbook, but I have no idea what it’s like).
I was able to pass a full-length practice N5 test pretty comfortably after completing MNN 1, and a full-length practice N4 test after completing MNN 2. But, as mentioned, I had no speaking practice whatsoever, haha, so if you’re trying to get placed in a class, speaking might be something you also need to prioritize beyond just JLPT proficiency.
But if you choose either MNN or Genki, you’ll likely have no trouble finding a tutor who could help you practice speaking alongside either textbook curriculum. For more obscure textbooks, it might be a bit harder finding a tutor who can work with them.
On level 38 you’ll learn the highly infrequent (~top 200) N5 word 一緒. Then you can finally start reading…
… because if you started reading any time before that you’ll already have learned it from exposure alone.
A for anyoneone that can find a higher frequency word at a higher level
I’m seconding the Bunpro recommendation. I think it helps with learning grammar as a beginner. I’m still a beginner myself, but I think doing N5 on Bunpro helped me understand basic sentences a lot better.
I’ll add that Bunpro also has vocabulary decks targeted for specific JLPT levels. In my opinion, it would take more than the one-month trial to get through all of the content in the N5 vocabulary deck (which would require about 40 new words a day), but adding 10 new words a day would allow the original poster to see all 1,100 words in the deck before October.