The Colin Howson book uses a notation like R(a,b,c) for the application of a predicate R to the arguments or terms a, b, c.
It employs the upper case letters A-Z, perhaps followed by subscripts, to be predicates, so, for example, R, S₁, T₁2 are all predicates.
The software supports this.
But the software makes an extension.
Often, when working informally, authors will write Red(x) to mean that the predicate Red is applied to the variable x.
Colin Howson,  Logic with trees Chapter 11
Set theory is an extensive topic introduced elsewhere. It can be written as a first order theory.
There is one axiom schema, Abstraction (or Comprehension), which can generate infinitely many axioms
Colin Howson,  Logic with trees Chapter 9 & 11
It is common in this setting (which is arithmetic) to use functional terms like s(x), s(1), s(0) to mean the successor of x, 1, and 0, respectively. Equally common is the notation x', 1', and 0' to mean the same thing. The latter is quicker and shorter (though not semi-nmemonic)-- we will use it here.
You need to know some propositional logic to be able to understand the tutorials to come. In particular, you need to know about the symbols used in propositional logic, truth tables, satisfiability, consistency, and semantic invalidity (by counter example). You do not need to know propositional rules of inference and derivations.
Howson  will give you enough background.
Alternatively you could look at the first five propositional tutorials in Easy Deriver
Colin Howson,  Logic with trees Chapter 2
A central use for Trees is to produce a counter example to an invalid argument. To do this, you construct a tree with a complete open branch. You will be able to do this for invalid arguments (but not valid ones). Then you run up that branch assigning all atomic formulas True and all negations of atomic formulas False.
This software will let you try a few.