What are the essentials about classes I assume that every student of mathematics has a satisfactory intuitive understanding of set theory. By "satisfactory" I mean mastery of the concepts typically found in introductory chapters of many algebra or analysis textbooks. Now, when beginning the study of category theory one is forced to work with classes. One is usually motivated through the Russel's Paradox but very little further explanation is given. My question is:
What are the essentials that one should know in order to work with classes at the level of, say, graduate abstract algebra?
More specific questions: can we safely use notation like $\in$, $\subset$, $\cup$, $\cap$ etc? What about cartesian product of classes?
 A: Yes, this is a known problem of many expositions to higher mathematics. They don't put a lot of emphasis on the set theoretic background, leaving many students (which later become professors) feeling that set theory is perhaps inadequate somehow. Of course, this is done because set theory isn't the topic of a book about category theory. But still...
One simple way of thinking about classes is thinking about them as formulas. Namely, a class is given by a formula which defines it. So when we write $\alpha\in\mathrm{Ord}$, we mean that $\alpha$ satisfies the formula "$x$ is an ordinal". 
Similarly, when we talk about $x\in A\cup B$, this means that there are formulas $\varphi_A$ and $\varphi_B$ which define $A$ and $B$ respectively, and $x$ satisfies $\varphi_A\lor\varphi_B$.
This lets us apply all the rudimentary tools of the very basic set theory you mention to classes. With the one major exception: a class that is itself an element of another class, is a set. Proper classes are not "objects of the universe of sets", so they don't get to be elements.
You could argue that $V\in\{V\}$ can be written and it makes sense. But it just proves that $\{V\}$ is not a class.
