Mid-infrared (MIR) microcombs exhibit remarkable advantages for trace molecule detection, facilitating fast and precise spectral analysis. However, due to limitations in tunability and size of available MIR pump sources, it is difficult to achieve compact MIR mode-locked microcombs using traditional methods. Here, we propose the turnkey generation of MIR soliton and near-infrared second-harmonic microcombs in a single microresonator. The self-starting method based on self-injection locking eliminates the complex tuning process and is compatible with modular packaged MIR distributed-feedback lasers. By simultaneously leveraging second- and third-order nonlinearities, frequency up-conversion can be achieved without relying on external nonlinear media, facilitating the integration of f-2f self-referencing. Additionally, the direct turnkey generation of breathing solitons and two-soliton states has been achieved. Meanwhile, a new method is proposed that allows for quickly switching between different soliton states by simply changing the pump power. We provide operational guidelines and explain its operating mechanism. Dynamical analysis shows that the detuning automatically converges to the red detuning region corresponding to the target soliton state. Our work provides a full-system miniaturized mode-locking scheme for MIR microcombs and offers new potential for fast switching of soliton states and self-referencing of single-resonator.