This is a good question, but not one with a simple answer. The question also has no generic answer because there are many different and inequivalent statements of Newton's laws floating around. Mach 1893 was the big step forward in clearing up the interpretation of Newton's laws. As far as I can tell, modern textbook presentations tend to take a little of Newton's original formulation and mix in a little Mach.
If you look at Newton's original formulation of the laws, in context, it's pretty clear that although he was self-consciously imitating the style of Euclid, he wasn't actually doing that style of axiomatic reasoning. The first law is written as a special case of the first, which you wouldn't do if you really had in mind a formal axiomatic system. The examples he gives of the first law are not in fact valid examples of the first law.
Based on Newton's original formulation, and ignoring the important foundational issues pointed out by Mach, the first law is a consequence of the second. Conservation of momentum is a consequence of the third law. Conservation of mass is logically independent. Conservation of energy is logically independent (since there is nothing in the laws of motion that restricts fields of force to being conservative). Conservation of angular momentum is logically independent (since there is nothing in the laws of motion that prohibits forces that act at a distance and along directions different from the radius vector).
Going in the opposite direction, Newton's laws can all be proved from conservation of energy and momentum (or from conservation of energy and Galilean relativity). For a treatment in this style, see my book Simple Nature.
From a modern point of view, mechanics is only one tiny branch of physics, and the conservation laws are much more general than mechanics.
Mach, The Science of Mechanics, 1893; 1902 English Edition, tr. McCormack, http://www.archive.org/details/sciencemechanic00machgoog