We consider two dimensional Dirac fermions in the presence of three types of disorder: random scalar potential, random gauge potential, and random mass with long-range correlations decaying as a power law. Using various methods such as the self-consistent Born approximation (SCBA), renormalization group (RG), the matrix Green's function formalism, and bosonization, we calculate the density of states and study the full counting statistics of fermionic transport at lower energy. The SCBA and RG show that the random correlated scalar potentials generate an algebraically small energy scale below which the density of states saturates to a constant value. For the correlated random gauge potential, RG and bosonization calculations provide consistent behavior of the density of states, which diverges at zero energy in an integrable way. In the case of correlated random mass disorder, the RG flow has a nontrivial infrared stable fixed point leading to a universal power-law behavior of the density of states and also to universal transport properties. In contrast to the uncorrelated case, the correlated scalar potential and random mass disorders give rise to deviation from the pseudodiffusive transport already to lowest order in disorder strength.