One approach towards conics classification in hyperbolic geometry is to start with Bertrami-Klein model, place a normal Euclidean conic in there and classify it according to the intersections with the absolute. This leads to a handful of nice curves and for the most part, I can imagine how they look.
For example, an elliptic parabola (one double contact with the absolute and two imaginary intersections) would look roughly like an ellipse near its vertex, and it would more and more approximate an equidistant curve as we go towards the infinity along it -- all its points would stay within a certain distance from its axis. Various hyperbola-like curves would have asymptotes so I guess that far enough from their "interesting regions" their curvature would change very slowly and they would look just like straight lines.
The one curve where I still have a problem is the "semi-circular parabola" -- a curve with triple contact and an intersection with the absolute. This curve has a single branch, but no axis -- since one end of the branch "ends" in an intersection with the absolute and the other one in a triple contact, I am not sure how this curve would actually look to an observer in the hyperbolic plane.