Why is the universe not uniform? From my understanding, the Big Bang theory says the universe expanded from a state of infinite density and pressure. I feel that if this is true, this should lead to a universe where matter is distributed uniformly across space. Yet most matter is clumped up into stars and planets, with areas of nothing in between. What could have led to such an irregular universe? Did I make a wrong assumption somewhere?
 A: A smooth distribution of matter is an unstable equilibrium. By this I mean that if it is perfectly smooth then it remains stable, but if you introduce even the slightest deviation from smoothness then that deviation will increase with time. This is because grouping the matter together into clumps reduces its energy due to the decrease in gravitational potential energy.
We tend to assume that the universe started out perfectly smooth just after the Big Bang, though actually this is just an assumption used to keeps the maths easy and there is no special reason for us to make this assumption. Anyhow let's stick with this for now, in which case the question is how did the initially perfectly smooth distribution of matter develop deviations from smoothness.
The answer is that in the first $10^{-23}$ or so of a second after the Big Bang the universe went through a process of exponential expansion called inflation. This was driven by a field called the inflaton field, and during inflation quantum fluctuations in the inflaton field got magnified up to cosmological scales. These are known as primordial fluctuations and they provided the initial deviations from smoothness that subsequently grew into the structures we see today.
A: Implicit in the proposal of smoothness

I feel that if this is true, this should lead to a universe where matter is distributed uniformly across space

is the assumption of an equilibrium, a thermodynamic equilibrium is how smoothness is created in matter as we know it. In order for such an equilibrium to exist all the space time points  of the tiny universe immediately after the the Big Bang should be able to interact with each other. This is not true because in the model special relativity still holds and there are parts of the universe that do not have access to others, due to the light cones. 
Despite the above argument, the Cosmic Microwave Background data  show remarkable uniformity from the time that the photons decoupled  from the hadronic soup.

at the level of 10^-8  the universe showed uniformity that could not  have been achieved if General Relativity and Special Relativity, foundation stones of the Big Bang model, hold.
Looking at the details of the map in such definition one sees the blobs and depletions which led to the currently granular nature of the observable universe.

To explain the inconsistency, an effective quantization model for gravity was introduced for the first times after the Big Bang, before 10^-32 seconds . The inflaton with its quantum mechanical indeterminancy is not constrained by the particle definitions and the velocity of light, and thus it could churn the early universe into a homogeneous soup, the quantum mechanical fluctuations giving the observed inhomogeneity over the largely homogeneous early universe. 

Yet most matter is clumped up into stars and planets, with areas of nothing in between. What could have led to such an irregular universe?

Inflation has been continuing ever since 10^-32 seconds , space expanding as the Big Bang models currently, and this expansion has distanced the clumps generated by the quantum effects of the inflaton into what is currently a clumpy universe.

A: The reason are the tiny quantum fluctuations which are present to all times and cannot be "switched off". Those fluctuations have been amplified by gravity resulting in structure formation, that is formation of galaxies, stars, plantets, etc.
