In this work, we study the real-time monitoring of an information source over a communication link with transmission cost. As the most basic nontrivial setting, we consider a single source governed by a random walk process and design the scheduling policy that minimizes the cost sum of the transmission costs and the inaccuracy of the state information at the monitoring center. This problem closely relates to the emerging area of Age-of-Information (AoI) scheduling that focuses on the timely transfer of fresh information over resource-constrained networks. When the information updates are controlled by the scheduler, we can incorporate the internal dynamics of the information source into the transmission decisions. For a general cost function of estimation errors, we first establish the optimality of a threshold rule for minimizing a weighted sum of the transmission cost and the tracking inaccuracy for the information source. Then, we characterize the optimal threshold level as an explicit function of the transmission cost and the source behavior. Further, we provide preliminary results of a learning-based approach that finds the optimal threshold when the transmission cost is unknown a priori.