Using Monte Carlo dynamics and the Monte Carlo histogram method, the simple three-dimensional 27 monomer lattice copolymer is examined in depth. The thermodynamic properties of various sequences are examined contrasting the behavior of good and poor folding sequences. The good (fast folding) sequences have sharp well-defined thermodynamic transitions while the slow folding sequences have broad ones. We find two independent transitions : a collapse transition to compact states and a folding transition from compact states to the native state. The collapse transition is second-order-like, while folding is first-order-like. The system is also studied as a function of the energy parameters. In particular, as the average energetic drive toward compactness is reduced, the two transitions approach each other. At zero average drive, collapse and folding occur almost simultaneously; i.e., the chain collapses directly into the native state. At a specific value of this energy drive the folding temperature falls below the glass point, indicating that the chain is now trapped in local minimum. By varying one parameter in this simple model, we obtain a diverse array of behaviors which may be useful in understanding the different folding properties of various proteins.