Change appearance of ConvexHullMesh

While the answers provided so far are nice, it seems like there should be easier ways to achieve this. And there are, I will show two ways: 1) Keeping your ConvexHullMesh as a BoundaryMeshRegion object and (2 converting to a Graphics object.

SeedRandom[0]
pts = RandomReal[4, {200, 3}];
chull = ConvexHullMesh[pts];

First we use HighlightMesh with no need to recreate the MeshRegion:

HighlightMesh[chull, {Style[0, Directive[PointSize[0.015], Red]], 
  Style[1, Thin, Blue], Style[2, Opacity[0.5], Yellow]}]

Note the use of Style to style the various dimensions (0 for vertices, 1 for edges and 2 for facets. Also note that this object is still a BoundaryMeshRegion and you can compute other nice mesh-related properties from it.

Now the Graphics object approach.

Graphics3D[GraphicsComplex[MeshCoordinates[chull], {PointSize[0.02], Red,       
  MeshCells[chull, 0], Blue, MeshCells[chull, 1], Green, MeshCells[chull, 2]}], 
  Boxed -> False]

Note the use of the already available properties of the MeshRegion. No need to use Show here.

Show allows you to convert to Graphics object:

Graphics[Show[ConvexHullMesh[p]][[1]] /. {Directive[x_] :> 
    Directive[{Red, EdgeForm[{Black, Thickness[0.02]}]}]}]

Or perhaps a little more interesting:

pts = RandomReal[{0, 1}, {50, 2}];
g = Graphics[{Red, PointSize[0.03], Point[pts]}];
ch = Show[ConvexHullMesh[pts]][[1]] /. {Directive[x_] :> 
     Directive[{Yellow, EdgeForm[{Black, Thickness[0.02]}]}]};
Show[Graphics[ch], g]

Why the above solution (f@@q) doesn't work?

Because a MeshRegion is an AtomQ.

How do I change the appearance of ConvexHullMesh?

Because

The convex hull region is the smallest convex region that includes the points

the computed and returned ConvexHullMesh is of course correct. Because you are computing the convex hull, maybe it's because you are interested to show the input list of points and their convex hull. You can use Show to combine directly a MeshRegion with another Graphics.

Maybe you are interested to DelaunayMesh, related with but different from ConvexHullMesh (returns a MeshRegion instead of a BoundaryMeshRegion).

p = {{0, 0}, {1, 0}, {1/2, 1/2}, {1, 1}, {0, 1}};
q = ConvexHullMesh[p];
{Show[q, Graphics@Point@p], HighlightMesh[DelaunayMesh[p], Style[0, Red]]}