You could start from the premise that there was no net angular momentum in the universe at all; but it would still be the case that everything of interest was spinning.
On the scales of stars and planets there are (at least) two important mechanisms that result in individual systems having angular momentum. The first is turbulence. If you take a parcel of turbulent gas from a giant molecular cloud it will always possess some angular momentum, even if the total cloud does not. As the parcel collapses to form a star/planets conservation of angular momentum $J$ and dissipative interactions result in an ncrease in spin rate and collapse towards a planar geometry.
Second, stars form in clusters. There is interaction between stellar systems early in their lives. Again, the cluster may have little net J, but groups of stars can, relative to their own centre of mass frame.
On bigger scales (galaxies) the second of these explanations becomes more important. The interaction and accretion of galaxies is what gives individual galaxies a spin, even if the clusters they are born in have much less or even no net angular momentum.
As an example of how turbulent velocity fields lead to gravitational condensations containing angular momentum you could do worse than study the star formation simulation performed by Matthew Bate and collaborators. These simulations start off in clouds with zero net angular momentum, yet produce a host of stars with swirling accretion disks, binary systems of all shape and sizes etc. An example journal paper can be found here: http://adsabs.harvard.edu/abs/2009MNRAS.392..590B Here is a web page where you can download the animations and study them at length http://www.astro.ex.ac.uk/people/mbate/Cluster/cluster500RT.html
Turbulent clouds are by their nature random and stochastic in terms of their motions. Often the velocity field is defined in terms of a power law dependence on spatial scale. The formation of vortices is a characteristic of turbulent media. They can be produced in the absence of external forces. The vortices contain angular momentum.
It is also worth noting that not all galaxies have an appreciable spin. Spiral galaxies do, but many elliptical galaxies have little net rotation. See https://physics.stackexchange.com/questions/93830/why-the-galaxies-forms-2d-plane-or-spiral-like-instead-of-3d-ball-or-spherica