Wood – When sanding spindles how far should I go

The short ends you pictured are the cross grains of the board. What I see is that you have used a too coarse sandpaper on these edges. This should be easy to fix. You need to step down the grit of the sandpaper. Any final sanding should be done with 220 or 320 paper. On these end details, use fine paper, break its (paper's) back so it is flexible, and take your time working it on these ends by hand. It may take a few minutes, but it will smooth out nicely.

It is hard to tell from the picture but I am guessing you are using a traditional screen and you are moving the screen back and forth along one of the axes of the screen. If the less than stellar ascii art below is your screen, instead of instead of moving N-S or E-W move the screen NE-SW or NW-SE. Or start with the screen rotated 45 degrees and continue to sand N-S or E-W

+--+--+--+--+--+--+--+
|  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+          N
|  |  |  |  |  |  |  |        W + E
+--+--+--+--+--+--+--+          S
|  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+

A lot of times primer and paint will fill in these tiny scratches and most people will not notice even if the scratches are not filled in completely. But since it is your house and the scratches bother you go ahead and try the diagonal movements.