The propagation and interactions of Airy-Gaussian beams in a saturable nonlinear medium are investigated numerically based on the split-step Fourier transform method. We show that the propagation of a single Airy-Gaussian beam in the saturable nonlinear medium can generate breathing solitons under steady state conditions. The generation and propagation of these breathing solitons can be affected by the initial amplitude and the field distribution factor of the single Airy-Gaussian beam. In a certain power range, these breathing solitons propagate along the acceleration direction with a controllable tilted angle. In the domain existing in these breathing solitons and for a given value of the field distribution factor of the single Airy-Gaussian beam, when the initial amplitude of the single Airy-Gaussian beam increases gradually, the periodicity of these breathing solitons becomes from small to larger and the tilted angle of these breathing solitons increases monotonically. When the value of the initial amplitude of the single Airy-Gaussian beam is given, the bigger the value of the field distribution factor of the single Airy-Gaussian beam, the smaller the tilted angle of these breathing solitons. Furthermore, the stability of these breathing solitons has been investigated by using the beam propagation method, and it has been found that they are stable. We find that the propagations of two Airy-Gaussian beams in the saturable nonlinear medium can generate not only soliton pairs but also interactions between two Airy-Gaussian beams. When the two Airy-Gaussian beams interact with each other, it is found that the in-phase Airy-Gaussian beams attract each other and exhibit a single breathing soliton with strong intensity in the beam center and some symmetric soliton pairs with weak intensity near both sides of the beam center. The smaller the interval between the two incident Airy-Gaussian optical components, the stronger the attraction between two Airy-Gaussian beams, and the less the numbers of the soliton pairs. The energies of both the main lobes of two Airy-Gaussian beams and the single breathing soliton increase with the value of the field distribution factor of two Airy-Gaussian beams. On the other hand, the out-of-phase Airy-Gaussian beams repel each other and exhibit only symmetric soliton pairs on both sides of the beam center. Our analysis indicates that the repellant of two out-of-phase Airy-Gaussian beams becomes big when the interval between two incident Airy-Gaussian optical components decreases and the number of the soliton pairs becomes less when the field distributions of two beams are close to the Gaussian distribution.