Using different style on segments of a plot

MeshShading

Plot[Sin[x], {x, 0, 2 Pi}, MeshFunctions -> {# &}, 
 Mesh -> {{Pi/2}}, MeshShading -> {Red, Directive[Dashed, Blue]}, 
 PlotStyle -> Thick]

Two piecewise functions

Plot[{ConditionalExpression[Sin[x], x <= Pi], 
  ConditionalExpression[Sin[x], x >= Pi]}, {x, 0, 2 Pi}, 
 PlotStyle -> {Directive[Thick, Red], Directive[Thick, Dashed, Blue]}]

Post - processing

Plot[Sin[x], {x, 0, 2 Pi}, PlotStyle -> Directive[Thick, Red]] /. 
 Line[x_] :> (With[{xx = (Pick[x, #[[1]] <= Pi & /@ x, #] & /@ {True, 
         False})}, {Line@xx[[1]], Dashed, Blue, Line@xx[[2]]}])

ColorFunction

Plot[Sin[x], {x, 0, 2 Pi}, ColorFunctionScaling -> False, 
 ColorFunction -> Function[{x, y}, If[x <= Pi, Red, Blue]], 
 PlotStyle -> Thick]

One more trick is to use Show

Show[Plot[Sin[x], {x, 0, Pi}, PlotStyle -> Red], 
 Plot[Sin[x], {x, Pi, 2 Pi}], PlotRange -> {{0, 2 Pi}, Automatic}]