After learning about the fundamental group, and proving that $\mathbb{R}^n$ minus any countable set is pathconnected, I started wondering if the fundamental group of $\mathbb{R}^2\mathbb{Q}^2$ is known. Does anyone know whether or not it is? Or how one might go about determining it?

The space $X = \mathbb{R}^2  \mathbb{Q}^2$ is not semilocally simply connected, and so in a sense the fundamental group is a poor measure of the homotopy 1type of $X$. It is an exercise in Hatcher's book Algebraic Topology that this group is uncountable. Here is an attempt to motivate a different view of the fundamental group: Consider the space $X_{n,m}$ which is the complement in $\mathbb{R}^2$ of all points with coordinates $(p/n,q/m)$ where $n,m$ are positive integers and $q,p$ are integers. This forms a cofiltered diagram $\mathbb{N}^2 \to Top$ whose limit is $X$, and where each space in the diagram is semilocally simplyconnected. There is a map $X_{n,m} \to X_{n',m'}$ when $n'n$ and $m'm$. The fundamental group of $X_{n,m}$ is free on a countable number of generators (a generator is given by a small circle around each deleted point, and connect this circle to the basepoint, $(\pi,\pi)$, say, by a path), and the maps between them kill off certain generators. Already, you can see how complicated $\pi_1(X)$ is going to be, since it has a countable number of quotients, each of which is free on a countable number of generators (actually, each of them is isomorphic to the free group on generators $\mathbb{Z}\times\mathbb{Z}$). This system of groups is known as a (strict) progroup, and can be thought of as a formal limit This is usually the 'right' algebraic object to consider when one has a badly behaved space such as this one. It is this object which 'controls' what the covering spaces of $X$ look like, as $X$ doesn't have a universal covering space (which if it did exist, would have fibre $\pi_1(X)$, and other connected covering spaces would be quotients of this one). This gets into the realm of shape theory (wikipedia, nLab), developed to consider spaces with bad local homotopy properties. 

