A coarse-grained model of a self-avoiding tethered membrane with hexagonal coordination, embedded in three-dimensional space, is studied by means of extensive Monte Carlo computer simulations. The simulations are performed at various temperatures for membranes with linear size 5< or =L< or =50. We find that the membrane undergoes several folding transitions from a high-temperature flat phase to multiple-folded structure as the temperature is steadily decreased. Using a suitable order parameter and finite size scaling analysis, these phase transitions are shown to be of first order. The equilibrium shape of the membranes is analyzed by calculating the eigenvalues lambda(max) (2)> or =lambda(med) (2)> or =lambda(min) (2) of the inertia tensor. We present a systematic finite size scaling analysis of the radius of gyration and the eigenvalues of the inertia tensor at different phases of the observed folding transitions. In the high-temperature flat phase, the radius of gyration R(g) grows with the linear size of the membrane L as R(g) proportional to L(nu), where the exponent nu is approximately equal to 1.0. The eigenvalues of the inertia tensor scale as lambda(max) proportional to lambda(med) proportional to L(nu) and lambda(min) proportional to L(nu(min) ), whereby the roughness exponent nu(min) is approximately equal to 0.7. We also find that the time tau(R) of a self-avoiding membrane to diffuse a distance R(g) scales as tau(R) proportional to L(2nu+2), which is in good agreement with the theoretical predictions.