The idea of these is to make sure you understand them, not memorize them, and that you begin to think in terms of them when you have arguments or statements that they represent or that are something like them.  Using them incorrectly makes for common fallacies, so understanding these inferences and practicing with them a bit is one way of getting yourself to recognize some fallacies fairly easily.  There are other fallacies as well, but the ones below are typical inferences people make, and sometimes get wrong.

 I am using "--->" to mean "implies".  There are a few different ways that "implies" can be translated into English and vice versa.  So that "A implies B" or "A --> B" can be thought of as any of the following:

The symbol "--->" sometimes means other things in other contexts, even other logic contexts, so don't think the way I am using it below is universal.  This is just for symbolic ease here.

In any implication, the part of the "if" clause is called the "antecedent", and the part in the "then" clause is called the "consequent".  So, below,  when you have A ---> B, A is the antecedent and B is the consequent.

Some of the following may sound complicated, but it is pretty easy if you just think about them and work with some simple examples.
 

Modus Ponens (Affirming the Antecedent) If A implies B, and A is true, then B is true. A ---> B A______ B Another way to think of this is that if all A's are B's and you know something is an A, then you know it must be a B. The Associated Fallacy is the fallacy of "affirming the consequent" and thinking that means the antecedent must be true also.  That is not necessarily true. If all dogs are animals, and Bessie is an animal, it is fallacious to conclude that Bessie must be a dog. This seems obvious, but there are lots of cases where people make this mistake.  In science courses, for example, you were probably taught that scientific theories are confirmed by showing that the predicted consequences of the theory actually occur.  That would be fallacious reasoning.  (See my essay about it if you are interested: Scientific Confirmation -- an Explanation)	Modus Tollens (Denying the Consequent) If A implies B, and B is not true, then A must not be true. A ---> B not B  not A If all A's are B's, and you have something that is not a B, then you also know it is not an A. The Associated Fallacy is the fallacy of "denying the antecedent" and thinking that means the consequent must also be false.  That is not necessarily true, for if all dogs are animals, and Bessie is not a dog, it is fallacious to conclude that Bessie must not be an animal.
If all A's are B's and all B's are C's, then all A's are C's. A ---> B B ---> C A ---> C	If P is true or Q is true (or both), then if P is not true, Q must be true. P or Q not P  Q
One dilemma form (affirming an antecedent): If A implies B, and C implies D, then, if either A or C is true, then either B or D must also be true. (A ---> B) and (C ---> D) A or C B or D	A second dilemma form (denying a consequent): If A implies B, and C implies D, then if either B or D is false, then A or C is false. (A ---> B) and (C ---> D) not B or not D not A or not C
Form One of De Morgan's Theorem: If it is not the case that both A and B are true, then either A is false or B is false (or both)	Form Two of De Morgan's Theorem: If it is not the case that either A or B is true, then A is not true and B is also not true.
Transposition, which is a statement of contraposition: If A implies B, then not-B implies not-A Or "if all A's are B's, then all non-B's are non-A's" A ---> B  not-B implies not-A	If P implies Q, then either P is not true or Q is.
There are special cases when A and B are equivalent or imply each other.  For if (A implies B) and (B implies A), then (not B implies not A) and (not A implies not B). Another way of saying these is that "A is true if and only if B is true."   An example is that if you stick a clean toothpick into a baking cake and it comes out clean, the cake is done but if it comes out dirty the cake is not done.  That is because "the cake is done if, and only if, the toothpick comes out clean".  Or another way to put that is: If the toothpick comes out clean, the cake is done, and if the toothpick comes out dirty, the cake is not done. In "if and only if statements", it is not a fallacy to affirm the antecedent by affirming the consequent and it is not a fallacy to deny the antecedent by denying the consequent.  The reason is that the antecedent of the "if" clause is the consequent of the "only if" clause. The above might sound complicated, but it is pretty easy if you just think about it a bit and work through some examples.	"A only if B" can be understood as stating that "A implies B" or, what is the same thing by Modus Tollens, that "If B is not true, then A is not true." That is why in the equivalence case in the left hand column, to say that A and B imply each other is to say that A <---> B, which is another way of writing: (A ---> B) and (B ---> A)

Again, remember, the idea is not to try to memorize this table, but to be able to work with various kinds of implication statements and know which ones are necessary and which ones are not, when you have time to think about them carefully.  Or to put it another way, to know what things imply what, and which things do not imply what. 

The idea is not to be able to do these automatically (for some are complex, and there are many more of these than are listed here, and there are different ways of stating many of them in English that tend to obscure them), but to be suspicious immediately of any argument in their form until you can think through it to make sure whether it is a good, valid argument or not.

There are also various ways to make diagrammatic depictions of these things.  Venn Diagrams for example are useful for this, and there are different ways of doing Venn Diagrams.  But Venn Diagrams are not the only kinds of diagrams you can make for yourself.  Anything that helps you visualize what a sentence or paragraph is about is helpful.

Finally, as you said, you cannot read these fast.  These take time to understand and get straight in your own mind and then make deductions about.  As you practice them you can do them somewhat faster, but new ones always cause problems and need to be thought about.  That is why when Dr. Blair asked me whether I wanted you to have the HiB flu shot to guard against meningitis of a certain sort, I had to think about it when he (and the pharmaceutical brochure) gave me the following evidence:
1) 95% of meningitis comes from the HiB flu virus.
2) the shot totally prevents HiB flu.

His implication was 3) that I should give you the shot.

But that sounded suspiciously like a fallacious argument I heard on Dragnet one night when the cops told some guy who said that smoking marijuana in his home was harmless.  They told him it was not, because 90% of all heroin users began with marijuana, so that showed marijuana use put you at a high risk for going on to using heroin.

Well, 100% of heroin users began life as babies, but that does not mean that there is a high risk of going on to be a heroin user if you were once a baby.  It is not the percentage of heroin users that began as marijuana users, but the percentage of marijuana users that become heroin users that is crucial for showing whether marijuana tends to lead to heroin use or not.

Similarly, the HiB flu and meningitis stats above do not tell what the odds are of getting the flu or of getting meningitis from the flu, so it might turn out there is a very low risk of getting that meningitis.  And the crucial thing about realizing that is that it then makes you wonder what the risk of serious side-effects of the shot are. So that if the risks from the shot are more probable than the risks of getting the disease without the shot, then the shot is not a good thing to do.  That is, in fact, why they do not give smallpox vaccinations any more to babies -- because the risk of the shots is far greater than the risk of smallpox at this time in the world.

So I called Jefferson County Health Department to find out what the risks of the shot and of getting meningitis were.  They did not know the incidence of HiB caused meningitis but they knew there were a few cases every year and that they were serious.  They did know that there were no reported side effects from the shot and that it was one of the safest vaccines ever developed.  So I had Steve give you the shot.  But it all took me time to sort out.  So don't try to rush these.  They are not usually automatic or easy without practice or without having seen a particular form before.

If you want to see another example of something weird, but interesting, about these kinds of things, see my essay "Confirmation by Contrapositive Instances".  This is a solution I offer to "Hempel's Paradox" -- that if "All swans are white" is the same as "All non-white things are non-swans" than not only should finding a swan that is white be a confirmation of "All swans are white" but so should be finding a red brick, since finding a red brick is to find a non-white thing that is also a non-swan.  But since red bricks don't confirm that all swans are white, how can "all swans are white" be equivalent to "all non-white things are non-swans"?  That is the paradox.