Chapter 1 of Hatcher corresponds to chapter 9 of Munkres. These topology video lectures (syllabus here) do chapters 2, 3 & 4 (topological space in terms of open sets, relating this to neighbourhoods, closed sets, limit points, interior, exterior, closure, boundary, denseness, base, subbase, constructions [subspace, product space, quotient space], continuity, connectedness, compactness, metric spaces, countability & separation) of Munkres before going on to do 9 straight away so you could take this as a guide to what you need to know from Munkres before doing Hatcher, however if you actually look at the subject you'll see chapter 4 of Munkres (questions of countability, separability, regularity & normality of spaces etc...) don't really appear in Hatcher apart from things on Hausdorff spaces which appear only as part of some exercises or in a few concepts tied up with manifolds (in other words, these concepts may be being implicitly assumed). Thus basing our judgement off of this we see that the first chapter of Naber is sufficient on these grounds... However you'd need the first 4 chapters of Lee's book to get this material in, & then skip to chapter 7 (with 5 & 6 of Lee relating to chapter 2 of Hatcher).
There's a crazy amount of abstract algebra involved in this subject (an introduction to which you'll find after lecture 25 in here) so I'd be equally worried about that if I didn't know much algebra.
These video lectures (syllabus here) follow Hatcher & I found the very little I've seen useful mainly for the motivation the guy gives. If you download the files & use a program like IrfanView to view the pictures as you watch the video on vlc player or whatever it's much more bearable since you can freeze the position of the screen on the board as you scroll through 200 + pictures.
I wouldn't recommend you treat point set topology as something one could just rush through, I did & suffered very badly for it...
It is definitely for someone who already knows the subject and is looking for a different perspective. It is not advanced and it is not introductory, more a supplement. It is also sloppy and very hard to follow for someone who does not know the subject. The praises are from people who know the subject and like the presentation and a few things not easily found elsewhere. Overall, it is not a good book in my opinion.
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You would definitely need some basic understanding of manifolds, but I don't think you will need too much. I think you will need definitions, closed submanifolds, immersions/submersions, tangent and cotangent bundles, vector fields and differential forms. Though the book defines differential forms, it chooses algebraic approach. So it would be difficult to read if you don't know what differential forms really are. But you don't need to read the whole book on manifolds. If you will need some extra-stuff, you can always look it up.
Loring Tu's book seems to be a bit too slow (at least for me). I would recommend Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups". You will only need Chapters 1 and 2 (except "Differential ideals"). Altogether it is only about 50 pages, and I think Warner gives a concise and clear introduction to the subject.
In general, I recommend after getting a bit comfortable with manifolds to start reading Bott-Tu. If you will see some unfamiliar term, you can always return back and learn about it. Otherwise you have a risk of spending too much time for learning a lot of things that you don't need for the book, and some of them might not be important at all for you in the near future.
Speaking about exercises in Bott-Tu, there are indeed not too many of them, and most of them are pretty easy. But surprisingly enough, even though I didn't solve tons of exercises, I have learned a lot, and I have gained a lot of skills from this book. So I personally don't think you will need some extra-book with exercises.
But if you will feel that you need some more problems on algebraic topology, there are a lot of nice books. I can recommend Hatcher's book (though it is again a bit too wordy in my opinion), or these notes by Sossinsky. They are not following Bott-Tu book, but there are a lot of common topics.