How best to prep a heavily varnished handrail

There is no need to sand all the varnish off . Get some 120 grit sandpaper and sand everything until it feels and looks smooth, Use some filler to repair any dings or dents and then give it all light sand at 240 grit, Wipe it all down with a tack rag and then apply paint

As RB Said if you just want to paint over there is no need to sand back to bare wood. thought where you have apply knotting solution where needed before priming and painting. If you do strip it with paint and varnish remover, it is a very messy job. make sure everything you want to keep is well away from the job. follow the instructions, but you will find it drys out quickly, try applying and wrapping segments in plastic to keep it moist and active for longer. Have good ventilation! Its a nasty long process. probably better off just keying the surface with 120 grit then priming with 2 coats (maybe more if you get a lot of bleed though) to make sure the top coat of paint sticks.