The emergence of local phases in a trapped two-component Fermi gas in an optical lattice is studied using quantum Monte Carlo simulations. We treat temperatures that are comparable to or lower than those presently achievable in experiments and large enough systems that both magnetic and paired phases can be detected by inspection of the behavior of suitable short-range correlations. We use the latter to suggest the interaction strength and temperature range at which experimental observation of incipient magnetism and d-wave pairing are more likely and evaluate the relation between entropy and temperature in two-dimensional confined fermionic systems.