Physical theories describe how things change in given circumstances (if the theory is right). In practice this means that they are applied in simplified ways, where the simplifications of circumstances are assumed to not be too severe so that the predictions will not have too large errors.
In Newtonian gravity two bodies orbit their center of mass in elliptic orbits. This is a valid simplification of how planets move around the sun that ignores the mutual gravitational influences of other planets: it becomes a bad approximation when the others get too close, have large masses, or if one let their influence last over very long timescales. Then a more complex model is needed for good predictions.
In the case of the Earth-Moon-Sun system the orbits are almost but not completely closed, yet it is rarely worth using the full three-body model in Newtonian mechanics (mainly since it doesn't have closed-form mathematical expressions): intermediate approximations are used to describe the actual motion to high precision.
Similarly in Einstein's general relativity two-body orbits are not ellipses and have a bit of precession (and, for bodies of comparable masses, we do not have closed-form expressions for them). This is again well modeled mathematically. One can also take gravitational wave emission into account, making them slowly spiral inwards. Here the mathematics of the underlying physical theory allows us to approximate how much this happens, and to numerically simulate the complex pattern of spacetime distortion.
So, do planets move in circles? To a first approximation, yes. But if you do careful astronomical observations, you will find them to move in ellipses (explained by Newtonian gravity). Even more detail, and you notice precession of these ellipses (due to oblateness, other planets, general relativity). Even more precision, and you will notice other gravitational deviation, as well as slow inspirals due to tidal forces and gravitational waves. These high precision details are small and hint at the complex interactions between parts of the world, even when underlying physical theories are known.