Short Answer
There is no royal road to a strong argument. It takes a lot of diligence and mastery of a number of subject matters including mastery of the facts of the domain of discourse, formal and informal logic, linguistics, argumentation theory, and having a good range of argumentation examples. Professional philosophers spend their whole lives trying to make good arguments, but it is possible to see what are relevant topics.
Long Answer
The first step to building strong arguments is to understand what the purpose of the argument is. Arguments might be for debate, cooperative education, entertainment, and so on. Who your audience is and why they participate makes a difference. Going back to at least Aristotle's Rhetoric, argumentation has been a central preoccupation of philosophers and the nature of argumentation. There are many conceptual frameworks regarding what factors make for a good argument, perhaps most famously: ethos, pathos, and logos. Most relevant to practical argumentation is a thorough understanding of formal and informal logic. Formalized logic comes in many flavors and over the last few hundred years been studied very symbolically. You mentioned conditional, contrapositive, etc. Absolutely if an opponent makes an error conflating the truth of the inverse/converse and the conditional/contrapositive, go after it. Generally speaking, these sorts of mistakes are known as formal fallacies.
Another aspect of argumentation is informal logic. Whereas formal logic is concerned with the syntax of argumentation, informal logic focuses on the semantics. In philosophy, of great historical importance is the linguistic turn which led to related schools of thinking about philosophy like ordinary language philosophy. One representative of that school in the analytic tradition is Stephen Toulmin who is known for the Toulmin model outlined in The Uses of Argument. Unlike formal fallacies, informal fallacies are much more difficult to spot and refute. It helps to memorize common forms of both formal and informal fallacies such as this list on WP.
Another important way of strengthening arguments is by understanding the different types of logics. Contrary to popular beliefs, there are many formal and informal logics: three famous logics are propositional, predicate, and Boolean. And there are also three generally recognized forms of inference: deduction, induction, and abduction. Knowing how to use these is of utmost importance, because arguments draw, by definition, inferences of one sort or another. Also, of relevance is the terminology and the methods of recognizing and addressing the differences between formal and informal inference.
A final way to strengthen arguments is to understand the relationship between words, truth, and meaning. This is a central aim of linguistics. Theories of semantics are manifold. My personal favorite is cognitive semantics because it extends an observation by Ludwig Wittgenstein about the similarity of family resemblances and word meanings. This observation has resulted in prototype theory from the pivotal linguistics of Eleanor Rosch.
If these ideas are new to you, then welcome to philosophy whose motto might be Onus probandi incumbit ei qui dicit, non ei qui negat!
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