This paper study a general nonlocal problem characterized by the equation:

$$-\mathcal{L}_K u+V(x)u=P(x)h(u)\quad \text{in} \ \ \mathbb{R}.$$

Here,  $\mathcal{L}_K$ represents a nonlocal integro-differential operator, and $h$ is a nonlinear term displaying subcritical  or critical growth of Trudinger-Moser type. We initially prove the existence of a ground state solution by employing variational  methods and innovative analytical approaches. Furthermore, through the application of constrained variational methods, Brouwer  degree theory, and a quantitative deformation lemma, we establish the existence of a sign-changing solution with minimal energy,  surpassing the energy of the ground state solution.