How to substitute the following conditions into an expression?

A reliable approach would use the third argument of Reduce as variables to eliminate (see Behavior of Reduce with variables as domain)

Reduce[{p == a b x + b^2 y + a c z, a b == 1, b^2 == 2, a c == 4}, {p}, {a, b, c}] 

p == x + 2 y + 4 z 

In the former editions of Mathematica (ver <= 4) Reduce used the third argument for eliminating another variables. It can be still used this way though it is not documented anymore, however its trace could be found in SystemOptions["ReduceOptions"].
If Reduce didn't work this way one would exploit Solve (it is still supposed to eliminate variables), e.g.:

Apply[ Equal, Solve[{p == a b x + b^2 y + a c z, a b == 1, b^2 == 2, a c == 4}, {p}, {a, b, c}], {2}][[1, 1]] 

or Eliminate:

Eliminate[{p == a b x + b^2 y + a c z, a b == 1, b^2 == 2, a c == 4}, {a, b, c}] // Reverse 

or even simply appropriate rules replacement (in general this approach cannot be used seamlessly though)

p == a b x + b^2 y + a c z /. {a b -> 1, b^2 -> 2, a c -> 4} // TraditionalForm 

Edit

It would be reasonable to mention another two functions useful in similar tasks. Taking this polynomial identiclly equal to zero:

poly = p - a b x - b^2 y - a c z; 

we can rewrite it in terms of another three polynomials which are also identically zeros by the assumptions:

{poly1, poly2, poly3} = {a b - 1, b^2 - 2, a c - 4}; 

thus we know that the resulting polynomial will be equal to zero as well:

Last @ PolynomialReduce[ poly, {poly1, poly2, poly3}, {a, b, c, p, x, y, z}] == 0 

p - x - 2 y - 4 z == 0 

similarily we can find a Groebner basis of polynomials { poly, poly1, poly2, poly3} eliminating unwanted variables {a, b, c}:

First @ GroebnerBasis[{ poly, poly1, poly2, poly3}, {x, y, z}, {a, b, c}, MonomialOrder -> EliminationOrder] == 0 

These two methods are more useful when we want to find different representations of (polynomial) expressions in polynomial rings, thus we needn't assume that polynomials { poly, poly1, poly2, poly3} identically vanish.