How to get the AMS mathbb font while using Palatino elsewhere

You have to restate the meaning of \mathbb:

\AtBeginDocument{
  \DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
  \DeclareSymbolFontAlphabet{\mathbb}{AMSb}}

This should go somewhere after loading mathpazo.

give the kpfonts a try, they are similiar to palatino, but have also sans serif and typewriter fonts:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{kpfonts}
\usepackage{amsmath,amssymb}

\begin{document}
\Huge
$\mathbb{F}\mathbb{D}\mathbb{G}\mathbb{H}$

\normalsize
Serif text, and
\textsf{Sans Serif}, and 
\texttt{Typewriter}
\end{document} 

There is in fact a package option for this: if you pass noBBpl to mathpazo, then its blackboard bold fonts are not loaded. That also means you have to load the AMS ones separately, but that just means using amssymb, which is simpler than defining another symbol font alphabet:

\documentclass{article}
% these two \usepackage commands make it work
\usepackage{amssymb}
\usepackage[noBBpl]{mathpazo}
\usepackage[scaled=.95]{helvet}
\usepackage{courier}
\begin{document}
In this document,
\begin{itemize}
    \item $\mathbb N$ will denote the natural numbers,
    \item $\mathbb Z$ will denote the integers,
    \item $\mathbb Q$ will denote the rational numbers,
    \item $\mathbb R$ will denote the real numbers, and
    \item $\mathbb C$ will denote the complex numbers.
\end{itemize}
\end{document}