Which constructions on a category are still interesting for a groupoid? By a groupoid, I mean a (small) category in which every morphism is an isomorphism. 
It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits become utterly uninteresting for a groupoid. On the other hand, the core of a category is a groupoid, and it is sufficient to known the allowed isomorphisms to get the core. But what else can I do with this groupoid than talking about isomorphic objects and studying the automorphism groups?

Background Defining the objects and isomorphism of a (otherwise partly undefined category) seems to be much easier than to nail down the (most) appropriate definition of (the allowed) morphisms for a problem at hand. (One reason for this is that I'm often not even sure whether the appropriate morphisms will be maps, or whether relations would be more appropriate.) For example, my preferred introductory "Mathematical Logic" books defines the isomorphisms between two $S$-structures on page 41 is full generality, but only defines the homomorphisms between two $S$-structures on page 204 (and this definition is only appropriate for universal-Horn structures).
 A: From the way I interpreted your question, my answer would be that I don't think there is anything interesting you can do with a connected groupoid other than looking at the size of the set of objects and the automorphism group of one of the objects. A connected groupoid is determined up to isomorphism by these two properties. Any other property you will study about them will be derived from these two properties.
See this answer: 
A comparison between the fundamental groupoid and the fundamental group
Of course for non-connected groupoids, you just need to study these two properties (automorphism group and size of set of objects) for every connected component of the groupoid.
A: The answer of Amr focuses on connected groupoids. In a certain sense, a connected groupoid is not significantly different from it's automorphism group. However, the very reason why groupoids are more fundamental than groups is that they can have more than one connected component.
One interesting construction enabled by having more than one component is to compute (formal) sums over certain "properties" of the different components. Before proceeding, notice that the connected components are exactly the isomorphism classes of the groupoid, which are also the isomorphism classes of any category with the same core.
Take for example the following strange complex characteristic function of category $C$:
$$\chi_C(z)=\sum_{\text{isomorphism classes [x] of C}} \frac{z^{|[x]|}}{|\operatorname{aut}(x)|}$$
where $|\operatorname{aut}(x)|$ is the cardinality of the automorphism group of the object $x$ and $|[x]|$ is the number of objects in the isomorphism class of object $x$. The value $\chi_C(1)$ is defined even in case that $|[x]|$ is undefined or uncountable, and I vaguely remember an old paper by John Baez (et. al) that said something interesting about $\chi_C(1)$.
I haven't yet studied groupoids in sufficient detail, but I guess there are many interesting constructions as soon as you stop focusing on connected groupoids.
