>>863
Amann Escher do both analysis and calculus, so putting it at the end of your list isn't really ideal imo. Shilov has no prerequisites either. I'd recommend reading both earlier/now instead. The first chapter of Amann Escher is also a nice introduction to proof-based math in general.
For algebra, I read Gaitsgory's Math122 and Math123 notes (both are on AMS) combined with Lang's Algebra (didn't complete it though). Gaitsgory's notes on Math122 can also be used as a somewhat abstract course in linear algebra. I think he actually based his Math55a course on them, of which you can also find notes online. I also read the first few chapters of Gorodentsev but can't say much about it as I never finished it.
For Topology, I really like Brown's book "Topology and Groupoids". It has few prerequisites (if any) and is very interesting as it emphasizes applications in differential topology, algebraic topology and algebraic geometry (for instance, it introduces the Zariski topology early on in an exercise) whereas e.g. Munkres is far too focused on point-set topology, which you'll find in analysis books like Amann Escher anyway.
If you want a book that treats the differential geometry stuff in amann escher 3 in more detail, I recommend Sternberg's lectures on differential geometry. Kobayashi-Nomizu is the go-to book if you want to specialize in it (note that it'd most likely be a very difficult first read).
If you're interested in geometry I also highly recommend learning physics.