Finite mixture models have long been used across a variety of fields in engineering and sciences. Recently there has been a great deal of interest in quantifying the convergence behavior of the \emph{mixing measure}, a fundamental object that encapsulates all unknown parameters in a mixture distribution. In this paper we propose a general framework for estimating the mixing measure arising in finite mixture models, which we term minimum $Φ$-distance estimators. We establish a general theory for the minimum $Φ$-distance estimator, where sharp probability bounds are obtained on the estimation error for the mixing measures in terms of the suprema of the associated empirical processes for a suitably chosen function class $Φ$. Our framework includes several existing and seemingly distinct estimation methods as special cases using a weakened identifiability condition, but also motivates new estimators. For instance, it extends the minimum Kolmogorov-Smirnov distance estimator to the multivariate setting, and it extends the method of moments to cover a broader family of probability kernels beyond the Gaussian. Moreover, it also includes methods that are applicable to complex (e.g., non-Euclidean) observation domains, using tools from reproducing kernel Hilbert spaces. It will be shown that under general conditions the methods achieve optimal rates of estimation under Wasserstein metrics in either minimax or pointwise sense of convergence; the latter case can be achieved when no upper bound on the finite number of components is given. Also of interest is a sharp inequality that captures the local information geometry for general mixture models precisely in terms of moment differences between mixing measures.