We study the folding dynamics of short two-dimensional self-avoiding lattice model proteins and copolymers with specific HP sequences (H, hydrophobic, P, polar) that fold to unique native structures. HH contacts are favorable. The conformational landscape is found by exhaustive enumeration using two different move sets, and time evolution is modeled by a transition matrix with the Metropolis criterion. The kinetic sequence of folding events depends strongly on both the monomer sequence and on the move set. But certain features are common to all sequences with both move sets. Beginning with open conformations, chains fold through multiple paths, are slowed by kinetic traps, and accumulate in compact nonnative conformations. Most chains must surmount energy barriers, which we identify as the transition state. Strikingly, the transition state involves a chain expansion, due to the breaking of HH contacts of compact denatured conformations. Mutations can significantly affect the energy landscape and folding dynamics, but in ways that appear too subtle to predict just from knowledge of native structures. Consistent with other models, we find that sequences with a large energy gap between the native and the next higher energy level can be stable and fold rapidly.