We tackle the problem of system identification, where we select inputs, observe the corresponding outputs from the true system, and optimize the parameters of our model to best fit the data. We propose a practical and computationally tractable methodology that is compatible with any system and parametric family of models. Our approach only requires input-output data from the system and first-order information of the model with respect to the parameters. Our approach consists of two modules. First, we formulate the problem of system identification from a Bayesian perspective and use a linear Gaussian model approximation to iteratively optimize the model's parameters. In each iteration, we propose to use the input-output data to tune the covariance of the linear Gaussian model. This online covariance calibration stabilizes fitting and signals model inaccuracy. Secondly, we define a Gaussian-based uncertainty measure for the model parameters, which we can then minimize with respect to the next selected input. We test our method with linear and nonlinear dynamics.