You have a vector-valued function to plot. You may try with VectorPlot3D or ListVectorPlot3D. Here is a custom plot using 3D graphic primitives:

er[r_, \[Theta]_, \[CurlyPhi]_] = 
  D[FromSphericalCoordinates[{r, \[Theta], \[CurlyPhi]}], r];
etheta[r_, \[Theta]_, \[CurlyPhi]_] = 
  D[FromSphericalCoordinates[{r, \[Theta], \[CurlyPhi]}], \[Theta]]/
   r;
ephi[r_, \[Theta]_, \[CurlyPhi]_] = 
  D[FromSphericalCoordinates[{r, \[Theta], \[CurlyPhi]}], \
\[CurlyPhi]]/(r*Abs[Sin[\[Theta]]]);
f[r_, \[Theta]_, \[CurlyPhi]_] = 
  Simplify[r*Cos[\[Phi]] er[r, \[Theta], \[CurlyPhi]] + 
    r*Sin[\[Phi]] ephi[r, \[Theta], \[CurlyPhi]] + 
    Sin[\[Theta]] etheta[r, \[Theta], \[CurlyPhi]]];
With[{r = 1, n = 10}, 
 Graphics3D[{Table[
    Evaluate[{Point[FromSphericalCoordinates[{r, \[Theta], \[Phi]}]], 
      Arrow[{FromSphericalCoordinates[{r, \[Theta], \[Phi]}], 
        FromSphericalCoordinates[{r, \[Theta], \[Phi]}] + .3 f[
           r, \[Theta], \[Phi]]}]}], {\[Theta], Pi/n, Pi - Pi/n, 
     Pi/n}, {\[Phi], 0, 2 Pi, Pi/n}], Opacity[.5], Sphere[]}]]