Since you write that you already have practical experience with partial derivatives, I don't think Chapter 9 of Rudin is that bad. Its main purpose is to put the basic theorems of differential calculus in a rigorous form, especially the implicit function theorem. However, I do find that the statement of the rank theorem is not the clearest in Rudin.
I agree that Chapter 10 isn't a good presentation of the material there. It would be much better if it included discussion of submanifolds of $R^n$. The treatment of multiple integrals is extremely skimpy (presumably because Rudin thought this was compensated for by Chapter 11).
Alternative sources for the material of Chapters 9 and 10 are:
Calculus on Manifolds by Spivak. A concise presentation of multivariable differential calculus, multiple integrals and differential forms. It has a bit of a do-it-yourself feeling to it, but it is far better than the later chapters of Rudin. (Submanifolds of $R^n$ only.)
Advanced Calculus by Loomis and Sternberg. Contains much more material than Spivak, on many topics. (Abstract manifolds.)
Mathematical Analysis II by Zorich. (Abstract manifolds.)
Differential Calculus and Differential Forms by Cartan. A very clean presentation of the material, but it assumes you have learned about multiple integrals elsewhere. (Submanifolds of $R^n$ only.)
Another way to approach this would be to learn the material on multivariable differential calculus and multiple integrals elsewhere (for instance in Chapters 12-14 of Apostol's Mathematical Analysis or in Chapters 7 and 8 of Burkill's Second Course in Mathematical Analysis) and to learn the manifolds/differential forms part within a more comprehensive book on manifolds. This could be economical if you plan to study differential geometry or topology anyway. One accessible book of this kind is Introduction to Manifolds by Tu (abstract manifolds). Another one with a different focus is Differential Topology by Guillemin and Pollack (submanifolds of $R^n$ only).
I think Chapter 11 of Rudin is really a different matter than Chapters 9 and 10. It's meant to provide a quick introduction to Lebesgue integration with a few of the basic convergence theorems. For that purpose, I don't think it's bad, although the presentation presupposes that you have read the excessively quick treatment of multiple integration (for continuous functions with compact support) in Chapter 10. On the other hand, many would prefer to learn this material as part of a broader and more complete introduction to the Lebesgue theory, such as that provided by standard textbooks like Rudin's Real and Complex Analysis or Lang's Real and Functional Analysis.
If you want alternative, light presentations of the Lebesgue integral, you could look at any of the follwing sources.
With measure theory:
Elements of Integration, Bartle.
Probability with Martingales, Williams (within the context of an introduction to measure-theoretic probability - it's best if you have studied elementary probability theory previously).
Without measure theory, restricted to $R^n$:
Mathematical Analysis, 2nd ed., Apostol (Chapters 10, 11, 15).
The Lebesgue Integral, Burkill.
If you happen to be really interested in probability theory, there's Billingsley's Probability and Measure as well, but this is quite lengthy.