How to replace $x$ and $x^2$ with different value?

Give the replacement rules in a list:

expr /. {x^2 -> k p + k (k - 1) p^2, x -> k p}

2 k p + (-1 + k) k p^2

If you are calculating the expectation of an expression, use Expectation or other built-in statistics functions rather than replacements which can be error-prone (which is why you ended up asking the question).

expr = x + x^2;

Expectation[expr, x \[Distributed] BinomialDistribution[k, p]] //
 Simplify

(* k p (2 + (-1 + k) p) *)

Alternatively, using TransformedDistribution and Mean

Mean[TransformedDistribution[expr, 
   x \[Distributed] BinomialDistribution[k, p]]] // Simplify

(* k p (2 + (-1 + k) p) *)

Verifying that both results are identical,

% === %%

(* True *)

This is entirely equivalent to Bob's answer:

x + x^2 /. x^m_. :> Moment[BinomialDistribution[k, p], m]
   2 k p - (1 - k) k p^2