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We consider the problem of minimizing upper bounds and maximizing lower bounds on information rates of stationary and ergodic discrete-time channels with memory. The channels we consider can have a finite number of states, such as partial response channels, or they can have an infinite state space, such as time-varying fading channels. We optimize recently proposed information rate bounds for such channels, which make use of auxiliary finite-state machine channels (FSMCs). Our main contribution in this paper is to provide iterative expectation-maximization (EM) type algorithms to optimize the parameters of the auxiliary FSMC to tighten these bounds. We provide an explicit, iterative algorithm that improves the upper bound at each iteration. We also provide an effective method for iteratively optimizing the lower bound. To demonstrate the effectiveness of our algorithms, we provide several examples of partial response and fading channels where the proposed optimization techniques significantly tighten the initial upper and lower bounds. Finally, we compare our results with results obtained by the conjugate gradient optimization algorithm and an improved variation of the simplex algorithm, called Soblex. While the computational complexities of our algorithms are similar to the conjugate gradient method and less than the Soblex algorithm, our algorithms robustly find the tightest bounds. Interestingly, from a channel coding/decoding perspective, optimizing the lower bound is related to increasing the achievable mismatched information rate, i.e., the information rate of a communication system where the decoder at the receiver is matched to the auxiliary channel, and not to the original channel.