In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel:
$K(x,y,t)=\frac{\exp(-\coth(2t)(x^2+y^2)/2-\operatorname{cosech}(2t)xy)}{\sqrt{2\pi\sinh(2t)}}$
Now if I have given some initial condition $\Psi(x,t=0)$ I should get the time evolution of $\Psi$ by convoluting $\Psi$ and $K$ (in an analogous matter as was done here for the heat equation: http://en.wikipedia.org/wiki/Heat_equation#Fundamental_solutions). But I am confused about the two spatial variables in $K$.
Can anybody tell me what Integral I actually have to solve to obtain $\Psi(x,t)$?
Your answer is, explicitly, in https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Natural_length_and_energy_scales in the first place.
The Mehler article you are in needs tweaking to imaginary time, so you solve the Schroedinger equation instead of the diffusion-type equation in that wiki. Besides, the proper link to https://en.wikipedia.org/wiki/Propagator#Basic_Examples:_Propagator_of_Free_Particle_and_Harmonic_Oscillator with the explanatory usage statement right before section 1.1 also gives you the same answer. Propagators propagate configurations at t=0 to such at arbitrary t, via the precise convolutions stated.