What determines my surface finish and how “nicely” my part turns out?

Speed, material, and cutting tools play into surface finish. Rotating the workpiece at a greater speed (rotational velocity) while decreasing the feed (translational velocity) will optimize surface finish, whereas low speed and high feed is better suited for roughing passes where efficiency, not smoothness, is key. Make sure your speeds and feeds are compatible with the material you are cutting! With your finishing passes, you would also benefit from using a cutting tool with a round nose, as this will more gently carve away at the material.