We study a goal-oriented communication system in which a source monitors an environment that evolves as a discrete-time, two-state Markov chain. At each time slot, a controller decides whether to sample the environment and if so whether to transmit a raw or processed sample, to the controller. Processing improves transmission reliability over an unreliable wireless channel, but incurs an additional cost. The objective is to minimize the long-term average age of information (AoI), subject to constraints on the costs incurred at the source and the cost of actuation error (CAE), a semantic metric that assigns different penalties to different actuation errors. Although reducing AoI can potentially help reduce CAE, optimizing AoI alone is insufficient, as it overlooks the evolution of the underlying process. For instance, faster source dynamics lead to higher CAE for the same average AoI, and different AoI trajectories can result in markedly different CAE under identical average AoI. To address this, we propose a stationary randomized policy that achieves an average AoI within a bounded multiplicative factor of the optimal among all feasible policies. Extensive numerical experiments are conducted to characterize system behavior under a range of parameters. These results offer insights into the feasibility of the optimization problem, the structure of near-optimal actions, and the fundamental trade-offs between AoI, CAE, and the costs involved.