Landau and Lifshitz or Goldstein classical mechanics For someone who has a PhD in electromagnetic engineering and a good mathematical background, and who wishes to move to physics and start by mastering classical mechanics, would it be better to read Landau & Lifshitz's book first or Goldstein's?  And although, ideally, reading both would be best, which would make a more complete reading (commensurate with modern education standards) on the subject and is more self-contained, if only one of them could be read (due to allowed time)? The intention is to have solid foundations in classical mechanics and then move on to other topics in physics (quantum mechanics, relativity, field theory, etc).
 A: Actually neither text has aged that well in view of the developments made possible by the availability of computing power.
L&L remains unique for its style, but very few amongst us mortals can easily follow the insights of Landau.
Strangely enough I think Goldstein has stifled the development (in the US) of more modern textbooks on the topic.
Although not ideal for a variety of reasons, a combination of Hand & Finch's Analytical Mechanics and Calkin's Lagrangian and Hamiltonian Mechanics might do the trick.
Hand & Finch is a bit scattered but it does contain some computer-based problems and it does discuss some of the more modern aspects of the field, like KAM theory etc.  Calkin is a smaller digest but does an extremely good job with Hamilton-Jacobi and related material, including KAM.  Both are at the level of Goldstein, less encyclopedic but much more modern.
There is another option, which is the text by Fetter & Walecka: Theoretical Mechanics of Particles and Continua.  The basic text is a bit old but it does have its moments, and the authors added to the basic text with an additional shorter supplement focused on non-linear mechanics.
A: Goldstein first.
L&L's Mechanics is for sure one of the most beautiful books ever written in physics. Every line you read you feel like reading a masterpiece. Some people complain that L&L misses the Noether's theorem. That is not quite true. In fact it does not mention it, but the whole book is written on basis of the relation between symmetry and conservation laws.
L&L is not a textbook in the sense it cannot be used alone to guide a throughout study on classical mechanics. The book assumes a good background on classical mechanics. It does not define elementary concepts. Moreover it does not follow a historical construction of classical mechanics, it simply starts with the Hamilton principle as a postulate. This is the book to read after you already have a good grasp of what classical mechanics is.
Goldstein is not as beautiful as L&L but it is a superb textbook. Everything is in there. It is self contained. It does follow a logical and historical construction, starting from the d'Alembert principle. It is a shame though that it does not obtain Hamilton Principle from d'Alembert Principle. It has very good examples and very good (and difficult) exercises. In my opinion is the best choice to use on a standard course on classical mechanics.
A: Goldstein, surely. L&L is great, but it is not self-contained and the exposition isn't that great (there's certainly a lot lost in translation). Actually, if you don't already have mastery over basic Lagrangian and Hamiltonian physics (for example, if you cannot derive the Euler-Lagrange equations from the principle of least action, or if you cannot perform a Legendre transformation on a given Hamiltonian), I recommend The Theoretical Minimum: What You Need to Know to Start Doing Physics. The first half of the book will be way too basic for someone who already knows Maxwell's equations, but the second half is perfect. Goldstein and L&L both overcomplicate things. Once you're confident with Hamiltonians, you can move straight to quantum mechanics, no need to dally on Goldstein.
Also, neither is a good introduction to special relativity - it's pretty much assumed you know it at an undergraduate level (why the "barn and pole paradox" and "twin paradox" are not paradoxes, how to do a Lorentz transformation, etc.). So you might want to supplement yourself with a quick perusal of Special Relativity (M.I.T. Introductory Physics Series), which is a first or second year undergraduate introduction to special relativity. Likewise the Feynman lectures recommended in the comments are first or second year undergraduate introductions to subjects, and maybe are not what you're looking for. Goldstein (chapter 7 if I recall correctly) then allows you to derive lots of great more difficult consequences of special relativity.
If you have exposure to Maxwell's equations, going to field theory won't be much of a problem, as you simply substitute "Lagrangian" for "Lagrangian density". Goldstein also covers this, and L&L's second book "Classical Field Theory" is great here, too. But again L&L isn't self-contained and the exposition can be confusing. 
tl;dr: Goldstein's book and both books by L&L are all worth having on your shelf. Both are at the introductory graduate physics level, and try to be self contained but practically rely on some exposure to hamiltonians and lagrangians, and mastery over special relativity. You do NOT need to touch (let alone master) these books to move on to nonrelativistic quantum mechanics. 
