Ballistic transport through a collection of quantum billiards in undoped graphene is studied analytically within the conformal mapping technique. The billiards show pseudodiffusive behavior, with the conductance equal to that of a classical conductor characterized by the conductivity sigma(0)=4e(2)/pi h and the Fano factor F=1/3. By shrinking at least one of the billiard openings, we observe a tunneling behavior, where the conductance shows a power-law decay with the system size, and the shot noise is Poissonian (F=1). In the crossover region between tunneling and pseudodiffusive regimes, the conductance G approximate to (1-F) x se(2)/h. The degeneracy s=8 for the Corbino disk, which preserves the full symmetry of the Dirac equation, s=4 for billiards bounded with smooth edges which break the symplectic symmetry, and s=2 when abrupt edges lead to strong intervalley scattering. An alternative, analytical, or numerical technique is utilized for each of the billiards to confirm the applicability of the conformal mapping for various boundary conditions.