For example if something was orbiting the sun only slightly further out than the earth, it would risk capture by the earth when the earth caught up to it in it's orbit. If it was orbiting at a great distance, then it would have an almost circular orbit around the barycenter of the sun-earth system (ignoring other planets for now). I understand the Lagrange points, but I'm wondering what happens when things are orbiting slightly further out.
This is called a "restricted three-body" problem. It is slightly simpler than a general three-body problem since one of the bodies can be assumed to be "light". However, even this simplified problem doesn't have an analytical solution.
The orbit of the satellite doesn't have a particular shape, and can be chaotic. In some situations it can be approximated usefully as perturbed ellipse: that is it can be approximated as a Keplerian orbit but with non-constant orbital parameters, which nevertheless vary in a fairly regular way: for example, the orientation of the ellipse might rotate, or precess, or the values of the inclination and eccentricity might oscillate.