So, we know that the gravity is responsible for pulling the Moon towards the Earth. But because it moves in an orbit, it makes me think that there must be a force that is causing the moon to travel in the direction other than the Earth's. So it doesn't fall straight to the Earth.
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Although the force is radial, the direction of motion is not the direction of the force, rather it is the direction of the velocity at any time $t$. In order to find out the dependence $\mathbf{v}(t)$ one must solve the equations of motion $\mathbf{F}(\mathbf{r}, \dot{\mathbf{r}})=m\mathbf{a}$.
Doing so with the gravitational potential $V(r) = -G\frac{mM}{r}$ gives back trajectories which happen to be conic sections. The only case when such trajectories can degenerate in straight lines is when the initial velocity is zero.
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