pts = RandomReal[{0, 1}, {19, 3}];
Show[ListPointPlot3D@pts, Graphics3D@Line@pts]

data = RandomFunction[RandomWalkProcess[0.3], {0, 10^2}, 3];
pts = Transpose@data["States"];
Show[ListPointPlot3D@#, Graphics3D@Line@#] &@pts

ListPointPlot3D[...]/. Point[a___] :> {Thick, Line[a]}

data = RandomFunction[RandomWalkProcess[0.3], {0, 10^2}, 3];
pts = Transpose@data["States"];
ListPointPlot3D[pts, ColorFunction -> "SouthwestColors"]

ptsT = Transpose[{Range[1, Length@pts], pts}]

func = Interpolation[ptsT];
ParametricPlot3D[func[[Tau]], {[Tau], 1, Length@pts},
 ColorFunction ->
  Function[{x, y, z, t}, ColorData["SouthwestColors"][z]]]