How can I find the roots for $x^4-18x^2-8x+21$ in a nice form?

Your equation seems to be a casus irreducibilis; the solutions cannot be expressed using only real numbers.

You can make them "nicer" by using Root objects (try RootReduce).

RootReduce@Solve[x^4 - 18 x^2 - 8 x + 21 == 0, x]

{{x -> Root[21 - 8 #1 - 18 #1^2 + #1^4 &, 1]}, {x -> Root[21 - 8 #1 - 18 #1^2 + #1^4 &, 2]}, {x -> Root[21 - 8 #1 - 18 #1^2 + #1^4 &, 3]}, {x -> Root[21 - 8 #1 - 18 #1^2 + #1^4 &, 4]}}

Or, you can use N and Chop to get arbitrary-precision real numbers.

Chop@N@Solve[x^4 - 18 x^2 - 8 x + 21 == 0, x]

{{x -> -3.80007}, {x -> -1.42412}, {x -> 0.896691}, {x -> 4.3275}}

To make the radical representation explicitly real, use the TargetFunctions option with ComplexExpand

roots = Solve[x^4 - 18 x^2 - 8 x + 21 == 0, x] // 
  ComplexExpand[#, TargetFunctions -> {Abs, Arg}] &

(*  {{x -> 
       -Sqrt[3 + 4*Cos[(1/3)*ArcTan[
                       Sqrt[39]/5]]] - 
         (1/2)*Sqrt[24 - 
               16*Cos[(1/3)*ArcTan[
                       Sqrt[39]/5]] - 
               8/Sqrt[3 + 4*Cos[(1/3)*
                         ArcTan[Sqrt[39]/5]]]]}, 
   {x -> 
       -Sqrt[3 + 4*Cos[(1/3)*ArcTan[
                       Sqrt[39]/5]]] + 
         (1/2)*Sqrt[24 - 
               16*Cos[(1/3)*ArcTan[
                       Sqrt[39]/5]] - 
               8/Sqrt[3 + 4*Cos[(1/3)*
                         ArcTan[Sqrt[39]/5]]]]}, 
   {x -> 
       Sqrt[3 + 4*Cos[(1/3)*ArcTan[
                     Sqrt[39]/5]]] - 
         (1/2)*Sqrt[24 - 
               16*Cos[(1/3)*ArcTan[
                       Sqrt[39]/5]] + 
               8/Sqrt[3 + 4*Cos[(1/3)*
                         ArcTan[Sqrt[39]/5]]]]}, 
   {x -> 
       Sqrt[3 + 4*Cos[(1/3)*ArcTan[
                     Sqrt[39]/5]]] + 
         (1/2)*Sqrt[24 - 
               16*Cos[(1/3)*ArcTan[
                       Sqrt[39]/5]] + 
               8/Sqrt[3 + 4*Cos[(1/3)*
                         ArcTan[Sqrt[39]/5]]]]}}  *)

roots // N

(*  {{x -> -3.80007}, {x -> -1.42412}, {x -> 0.896691}, {x -> 4.3275}}  *)

You can obtain Root objects with

Solve[x^4 - 18 x^2 - 8 x + 21 == 0, x] // FullSimplify

or

Solve[x^4 - 18 x^2 - 8 x + 21 == 0, x] // RootReduce

but numerical roots are perhaps easier to understand.

NRoots[x^4 - 18 x^2 - 8 x + 21 == 0, x]

Convert the output of NRoots with ToRules, and pick the real roots with Cases. Plot the equation with red real roots. I wish I had Manipulate when I was studying calculus.

Manipulate[
   Module[{r = Cases[x/.{ToRules[NRoots[x^4 + a x^2 + b x + c == 0, x]]},_Real]},
      Plot[
         x^4 + a x^2 + b x + c, {x, -5, 5},
         PlotLabel -> ToString@TraditionalForm[x^4+a x^2+b x+c]<>"=0\n roots \[Rule] "<>ToString[r],
         Epilog -> {Red, PointSize[0.02], Point[Thread[List[r, 0]]]},
         Frame -> True, BaseStyle -> {FontSize -> 16}, ImageSize -> 550]],
    {{a, -18.}, -20, 20, Appearance -> "Labeled"},
    {{b, -8.}, -20, 20, Appearance -> "Labeled"},
    {{c, 21.}, -100, 100, Appearance -> "Labeled"}]