Strong interactions between electrons are the foundation of a myriad of exciting phenomena in condensed matter physics. In this thesis, we investigated two such cases---topological antiferromagnetism in Mn$_3$X (X = Ge, Sn), and unconventional superconductivity in UTe$_2$–by measuring their elastic moduli via the resonant ultrasound spectroscopy and pulse-echo ultrasound techniques. Our measurements in Mn$_3$X find exceptionally large magnetoelastic coupling in the high-temperature antiferromagnetic state in both compounds, indicated by large derivatives of the Néel temperature with respect to hydrostatic pressure: $\left(39 \pm 3 \right)$ K/GPa in Mn$_3$Ge and $\left(14.3 \pm 2.0 \right)$ K/GPa in Mn$3$Sn. Our measurements also helped identify the antiferomagnetic phase in these compounds as a piezomagnetic phase, rare behavior in which the total magnetization depends linearly on applied strain. From our pulse-echo ultrasound measurements in UTe$2$, we conclude that its superconducting order parameter has only one component which is the same between samples showing one and two superconducting transitions. Our data further suggests that the order parameter transforms as the $B{2u}$ irreducible representation of the $D{2h}$ point group. Comparing RUS and pulse-echo ultrasound data, as well as considering high-energy X-ray diffraction microscopy data, we speculate that the origin of the two transitions seen in some samples of UTe$_2$ is due to structural inhomogeneity in these samples.