First, to dispel false conceptions:
Informal logic is not the contrary of formal logic, at least for some established meanings of 'informal logic'.
'Formal logic' is usually reserved for the formal study of truth-preserving inference, like deduction. But there's nothing preventing a formal study of not necessarily truth-preserving inference, so-called ampliative reasoning. Most argumentative patterns found in everyday reasoning are ampliative, in particular defeasible reasoning.
(Please note: Just because they are not deductive, it does not mean that they are never valid. Informal logic has had a particular interest for informal fallacies in the past (see below), which might have given the impression that the standpoint of informal logic is that every non-truth-preserving argument is a fallacy. This is certainly not the case. Informal logic is interested in what makes ampliative inferences context-dependently valid.)
Indeed, informal logic as a field has many contributions obtained using formal methods (see as a random example this application of Bayesian methods to everyday arguments). Also - I have never seen an explicit mention of this - the whole (highly formal) discussion in philosophy of science about the viability of inference to the best explanation is essentially a discussion pertaining to informal logic.
1. The practice of reasoning in theory
Informal logic is not a discipline, but draws from many disciplines to carry out and study very different tasks and topics, such as:
- competing definitions of “argument”
- argument identification
- burden of proof
- the empirical study of argument
- cognitive bias
- the history of argument analysis
- methods of argumentative investigation
- the role of emotion in argument
- argumentative exchange in different social contexts
(See SEP Entry on Informal Logic)
This seems a garden variety of questions and indeed most question might be legitimate topics in vastly different disciplines: cognitive bias -> psychology; argumentative exchange in different social contexts -> sociolinguistics; etc.
This thematic pluralism (and disunity) is not just due to the many approaches used to tackle given questions, but is a product of the very aim of informal logic, namely to develop assess and analyze arguments that occur in natural language discourse. Since natural language is used in vastly different contexts and in a variety of ways for a variety of aims, the studies tend to multiply and branch off in different directions. Researching 'the role of emotion in argument' may be vastly different when analyzing legal arguments of lawyers in front of a jury than when studying Euclid's diagrammatic reasoning.
In this sense, informal logic vastly fulfills your desideratum that
in natural reasoning, you would want many inference rules that correspond to the rules that are most common patterns of reasoning determined by some empirical method.
On the other hand, this kind of context-sensitivity gets in the way of studying common patterns across different domains, i.e. an effort at theoretical generalization, which you seem to be interested in. Countering this case-study trend, some parts of informal logic aim to construct a comprehensive account of these different types of argumentation. Historically, these attempts have mostly been made in a field called argumentation theory, a field drawing from logic and linguistics, which intersects with informal logic.
2. The practice of reasoning in practice
Your second goal, namely to "improve the [your?] practice of reasoning" is addressed by another major domain of informal logic called "critical thinking", another intersection where the tools of informal logic are applied with an inclusive educational goal in mind, not just limited to academe. You might have come across discussions about e.g. atheism referring to informal fallacies. This is a good example of notions pertaining to early phase of the field of informal reasoning, where a theory of fallacy was predominant, that slowly trickled down from the ivory towers of academe into pop culture and can be found today in every other Reddit thread. (This SE gets its own share by hosting many questions about fallacies by people otherwise not interested in philosophy.)
So, to answer your question: Informal logic is definitely the keyword you're looking for. The bad news is that the field is not systematized and so there is no simple "Introduction to informal logic" covering all aspects you might be looking for. The moral of the story here: Don't give up just because you don't find something fitting on your first try.
The best advice I can give is to have a look at the brand new Handbook of Argumentation Theory at your library, skim through the contributions, read what grasps your attention and follow the bibliographic references (and ask about them on Philosophy.SE!).
Classics in this field are (in chronological order):
- Stephen Toulmin: The uses of argument (1959) (see the Toulmin model)
- Chaim Perelman and Lucie Olbrechts-Tyteca: The New Rhetoric: A treatise on argumentation (1969)
- Frans Van Eemeren and Rob Grootendorst: A systematic theory of argumentation. The pragma-dialectic approach (2004) (see Pragma-Dialectics)
(IMO, the most interesting studies are not these big volumes - with the exception of Toulmin because of his clarity - but single papers produced in the field. So reading the Handbook might be a better idea than starting with these classics.)
In the "critical thinking" department the book Critical Thinking by Moore and Parker is a classic.
Historically, you might be interested in the "pragmatic logic" of the Polish School, which is customarily well known because of its key contributions to formal logic, less to informal logic. See
Marcin Koszowy, "Pragmatic logic and
the study of argumentation", 2010 (PDF).
Have a look at the SEP entry on Informal Logic
Personally speaking, a favorite application of informal logic of mine (in history of science) is Finocchiaro's analysis of Galileo's scientific arguments: Galileo and the Art of Reasoning: Rhetorical Foundations of Logic and Scientific Method (1980).