f[x_, y_, z_] := x^3 + y^2 - z^2
You can use the (afaik undocumented) option "Extrusion":
ContourPlot3D[f[x,y,z] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
"Extrusion" ->. 2]

You can also use Extrusion -> .2 or Method ->{"Extrusion" -> .2}.
Alternatively, use PlotTheme -> "ThickSurface":
ContourPlot3D[f[x, y, z] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
PlotTheme -> "ThickSurface"]

To make the mesh lines into Tubes you can post-process:
ContourPlot3D[f[x, y, z] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
ContourStyle -> Opacity[.5, Orange],
MeshStyle -> Red] /. Line[x_] :> Tube[x, .03]

or use (also undocumented, afaik) option setting Tube[radius] for MeshStyle:
ContourPlot3D[f[x, y, z] == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
ContourStyle -> Opacity[.5, Orange], MeshStyle -> Tube[.03]]

Note: The last two pictures are obtained using v9 (windows 10). Can't run the same code on v12 on free Wolfram Cloud because of cloud credit limits.
Line -> Tube[#, 0.1]orMeshStyle -> Tube[.1]& substitute desired radius for0.1? See https://mathematica.stackexchange.com/questions/54744/how-does-one-get-mesh-lines-at-0-in-parametricplot3d – Michael E2 Aug 11 '19 at 13:57