I'm in a similar position as well - perhaps at just a wee bit higher level (am learning grad-level math by myself). Below is a list of books that I have enjoyed. Some might perhaps be at a slightly higher level than you might need right now though.
1. Analysis - The Elias Stein, Rami Shakarchi Analysis series, Meassure Theory by Terrence Tao, Lebesgue Integration on Euclidean Spaces by Frank Jones, A radical approach to Lebesgue Integration by Bressoud
2. Algebra - Can't do much better than the classic texts by Herstein & Artin here. I know Artin doesn't get much love but after a point, you really begin to appreciate it.
3. Complex Analysis - is there a better math book than Visual Complex Analysis by Needham?
4. Topology - i'm partial to a Dover book on Topology by Theodore Gamelin. I learnt the basics of point-set topology & homotopy form this book. Great exercises.
5. Number Theory - check out the 2 problem books on number theory by Ram Murty. Melvyn Nathanson's book is quite good as well.
6. Combinatorics - Combinatorics of Finite Sets by Ian Anderson. Extremal Combinatorics by Jukna.
I'd agree with your assessment of Royden. Another fantastic book on Lebesgue Integration/Measure theory is Lebesgue Integration in Euclidean Spaces by Frank Jones. Fantastic textbook.
1. Analysis - The Elias Stein, Rami Shakarchi Analysis series, Meassure Theory by Terrence Tao, Lebesgue Integration on Euclidean Spaces by Frank Jones, A radical approach to Lebesgue Integration by Bressoud
2. Algebra - Can't do much better than the classic texts by Herstein & Artin here. I know Artin doesn't get much love but after a point, you really begin to appreciate it.
3. Complex Analysis - is there a better math book than Visual Complex Analysis by Needham?
4. Topology - i'm partial to a Dover book on Topology by Theodore Gamelin. I learnt the basics of point-set topology & homotopy form this book. Great exercises.
5. Number Theory - check out the 2 problem books on number theory by Ram Murty. Melvyn Nathanson's book is quite good as well.
6. Combinatorics - Combinatorics of Finite Sets by Ian Anderson. Extremal Combinatorics by Jukna.
7. Graph Theory - Diestel or Bondy/Murty.
8. Galois Theory - Rotman is great.