Golla-Hughes-McTavish (GHM) method has been shown to be an effective approach to model viscoelastic materials (VEMs). In the GHM model, a mini-oscillator has been used as the mechanical analogy to illustrate the relation of GHM parameters. However, the GHM mini-oscillators have not been studied in depth so far. In this paper, the damping and isolation characteristics of this two degree-of-freedom mini- oscillator are analyzed. Those characteristics are crucial to the effectiveness and limitations of the passive and hybrid (active-passive) vibration suppression techniques employing VEMs. Under harmonic excitations, the corresponding nondimensional relationships among parameters are derived. The transmissibility due to force and base excitations is investigated with respect to various parameters. The damping ability of the mini-oscillator is also evaluated. For those critical points and special cases, their conditions are identified and discussed. Several unique distinctions are observed when compared to previous studies on vibration absorbers and isolators. The analysis results of this research provide more understanding and physical insight to designers when considering VEM-based configurations including passive and hybrid systems for the purpose of vibration isolation and control.