I recommend Artin, if you are a strong student, interested in learning the material from a real master.
This book was written by Michael Artin, world famous algebraic geometer, and full professor at MIT, to teach a class of MIT sophomores, when he could not find a book he thought suitable. He taught from it for several years while polishing the notes.
It has also been used at many other schools at various levels, probably usually as a junior/senior level math major course, but it could well serve some beginning graduate students.
His point of view is that linear algebra is the most important subject in mathematics, which is surely true, so he teaches group theory based on (infinite) groups of matrices instead of (finite) permutation groups.
He ties them in closely with geometry, and number theory as well to a small degree.
Although there is little on number theory, what is there is not found in any other text I know of, i.e. Minkowski's finiteness of the class group of a quadratic number field, and ideal factorization in similar concrete cases.
The big advantage of this book is the global mastery of the author of many subjects touching algebra. The material literally rolls off his fingertips, and he has taken the trouble to make it understandable, with solid effort, by a student.
Artin is virtually the only elementary book to discuss lie groups, especially the basic examples of SU(2) and SO(3). He provides the necessary topology along the way.
When he discusses field theory he also mentions transcendental field extensions and gives a lovely short proof of the Hilbert nullstellensatz, also not at all standard in algebra texts.
He is not impressed with the importance of Galois theory so leaves it to the last chapter, where he nonetheless does a beautiful job on it.
The book I am familiar with by Hungerford, Algebra, was written as a graduate text, not an undergraduate text, so that could explain the steeper learning curve one reviewer noted. As graduate books go however, Hungerford's book has an extremely full set of proofs of almost all (apparently not quite all) theorems, and lots of problems. Indeed his motivation in writing the book was to provide a book the average grad student could learn from and could read, including the proofs.
It is intended specifically to provide background preparation for the PhD algebra exam at a typical grad school.
My objection to Hungerford is that I feel he gives very little insight as to why the theorems are true, he just plods through a proof. I.e. he does not teach me as much as Artin does. My students however have said it serves a useful alternate to more motivated treatments. He has lots of examples.
There may also be a lower level version of his book I am not familiar with.
Herstein, Topics in Algebra, was written as an honors sophomore algebra book for Columbia students, so it has lots of fun challenging exercises for bright young students. The exposition to me and many of my friends, is of the sort that looks easy but goes in one ear and out the other. Opposite to Artin it focuses on finite groups, but matrices are also treated.
One problem is that algebraists multiply their matrices in the opposite direction from the rest of the universe, so if you, learn it herstein's way you can only communicate with other algebraists at first, until you learn to turn it around.
I do not recommend it unless you just happen to like it. I am not familiar with the watered down version.
I think Artin might be a little hard for many students, but once you get ready for it, maybe by reading an easier number theory book like Dudley, it is the best book available for those who are ready for the level roughly between upper level undergrad and beginning grad study.
A nice book I think is well written and easier than Artin, is Abstract Algebra: a geometric approach, by Theodore Shifrin.
Neither Shifrin nor Artin touch on multilinear algebra, except for a little on quadratic forms and the spectral theorem, i.e. there are no tensor products, or alternating products. Hungerford covers those in detail, for grad students.
