What basic metaphysics should every philosophy student know?
The etymology of the term should suffice.
"Metaphysics" is not a term which either Plato or Aristotle used. It originated with Andronicus of Rhodes (~150CE). Lacking a significantly coherent statement in the beginning of the writings to work with, as would have been customary for a title in his day, Rhodes' organizing principle for the extant works of Aristotle (essentially categorizing a hodgepodge into a collection) was simply that they were placed on the shelf after the books on physics: ta meta ta physika biblia, i.e. "the books that come after the books on physics." Only later was this categorical placeholder naming considered taxonomic and thus began a pernicious history of metaphysicians soliciting agreement with weltanschauung and proferring the hermeneutical as if it were heuristic - all under the banner and misnomer of "philosophy".
See also: The Origin of "Metaphysics"
by Anton-Hermann Chroust
The Review of Metaphysics
Vol. 14, No. 4 (Jun., 1961), pp. 601-616
To be clear, philosophy - and this for 2500+ years - translates to "love of wisdom". "Love" in the context of initial utterance was akin to "respect", "virtue" or "reverence" - nothing to do with notions of romantic love common today. To the point, however, philosophy distinguished itself from sophistry and the professional counsel of wise people (think of lawyers, financial managers, doctors and such today) whose "wisdom" was often dispensed relative to the monetary or political circumstance available with its authority.
What then is wisdom apart from those deemed wise enough to dispense it? Some have said wisdom is the intelligent application of knowledge. This may be so, however, intelligent according to whom - and who would be the arbiter? No, wisdom simply obtains knowledge. And that is all. Philosophy is the virtue of obtaining knowledge; philosophy is respect and reverence for obtaining knowledge.
What then is knowledge? Knowledge is empirical verification of what is (else how do you "know" what is?) of which there are three kinds:
1) axiomatic, or self-evident knowledge, e.g. "2+2=4" or "all dividends require financing"
2) empirical knowledge, e.g. "brute" facts (pace Anscombe) such as "the Earth is ~93 million miles from the Sun" or "institutional" facts such as "Obama is President"
3) self-knowledge, e.g. "I feel glad"
At first glance, it might seem circular or redundant to consider "empirical knowledge" as "empirical(ly) empirical verification of what is" but I can only ask that you consider that it is just really, really basic. Think about it, eh? Is there any good reason to presume otherwise, much less entertain non-illuminating skepticism regarding the senses? It might wreak havoc with the outliers that we are all deceived by an "evil demon" but we did not build the internet by wondering if we are merely batteries for the matrix... So, ask yourself: do you "know" that Obama is President if you have not empirically verified the statement corresponding to what is the case in the world? Do you "know" that you feel fine if you have not empirically verified yourself feeling fine? Do you "know" 2+2=4 if you have not observed the calculation of, having done the calculation, what is a self-evidenced truth that 4=2+2 is 2+2=4? If you must, you can consider that which is observationally verifiable when it is verified by observation to be empirical knowledge: in short the world is that which is empirically verifiable (read: known).
Even so, is it possible for anyone to suggest some very basic concepts
(especially ones more relevant to Greek philosophy)? Or introductory
books on metaphysics?
The domains of philosophy are epistemology and ontology. In addition to studying logic, reason and rhetoric, if you are interested in the ancients, then you would do well to learn ancient Greek. Lastly, consider that it is logically impossible to know the past.