One of my favourite manga, and in fact the first I ever read in the original Japanese, is よつばと!/Yotsuba&!, by 東清彦/Kiyohiko Azuma. It is an entertaining and easygoing manga, with good art and story, so that alone is enough to recommend it; it is a best seller in Japan so many would agree. However, I think it also happens to be the ideal beginner’s manga for learning Japanese, for several compelling reasons.
Firstly, it has furigana, not an uncommon feature by any means – less time spent looking up kanji means more time to increase comprehension. More interestingly from the learner’s perspective, is that the protagonist, Yotsuba, is a young child who often has to have things explained to her in simple terms, and this tends to make the business of interpreting Japanese text without a side by side guide in another language much easier.
A very basic vocabulary and a knowledge of simple grammar, coupled with the ability to read kana and use a dictionary, should allow anyone to read and enjoy this, whilst improving their Japanese at the same time.
Textbooks tend to be a difficult route for maintaining motivation to learn (they are tedious, and they are after all merely a means to an end) and are often a little divorced from the language as it is actually used (although clearly manga usage can also fall into this category). For this reason, some sort of enjoyable pastime is best combined with learning, and manga fits the bill for Japanese (RPGs are my other recommendation).
Even if Yotsuba-to is not the kind of Japanese material you eventually see yourself reading/watching (in my case I worked towards the likes of Berserk and Ghost in the Shell), it certainly seems to provide a “real” entry point to the language, when understanding a lot of other works is likely to be very difficult.
Even if you’re not learning Japanese, Yotsuba-to is a widely licensed and scanlated manga, so language should not be a great barrier to enjoying it.