There are two books which link category theory and group theory via the use of groupoids. A pioneer in this was Philip Higgins, whose 1971 van Nostrand Notes are now available as free download as Categories and Groupoids (T&G). Following Philip's lead I started investigating groupoids in $1$-dimensional homotopy theory in 1965, and concluded that the groupoid point of view made many things clearer. The book I published in 1968 is now in its third edition and available as Topology and Groupoids. A key tool was the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, chosen according to the geometry, which dates from 1967. The above are the only undergraduate texts in English using this concept. here is a mathoverview discussion on many base points, and a presentation on the background in algebraic topology was given at Galway in December, 2014. See also my preprint page.
A feature of both of these books is the use of the following construction. Let $G$ be a groupoid with object set $X$ and let $f:X \to Y$ be a function to a set $Y$. Then we obtain a new groupoid which can be written $U_f(G)$ with object set $Y$ and a nice universal property. From this can be deduced the construction of coproducts of groups, and of free groups and free groupoids. This goes a long way to covering point 2 of @goblin's comment. Thus there is a combinatorial groupoid theory which in principle includes combinatorial group theory.
January 23, 2017 I should say that "combinatorial groupoid theory" is my own coinage, but also reflects JHC Whitehead's term "combinatorial homotopy theory".
The areas include those mentioned above and also fibrations and covering morphisms, groups acting on groupoids, (see T&G for all these). Other references are in my groupoids web page.