How do I change the color of itemize bullet? specific and default

A per-\item change of colour is possible using the optional argument of \item[..], and using the same label as itemize - $\bullet$ or \textbullet.

For changing the default settings, you can either supply an updated label to the itemize environment, or create your own using \newlist together with \setlist:

\documentclass{article}

\usepackage{enumitem,xcolor}

\setlength{\parindent}{0pt}% Just for this example

\begin{document}

A list with a specific \verb|\item| changed:
\begin{itemize}
  \item First item
  \item[\textcolor{blue}{\textbullet}] Second item
  \item Last item
\end{itemize}

Default \verb|itemize| list with updated \verb|label|:
\begin{itemize}[label=\textcolor{blue}{\textbullet}]
  \item First item
  \item Second item
  \item Last item
\end{itemize}

Completely new list \verb|coloritemize|:
\newlist{coloritemize}{itemize}{1}
\setlist[coloritemize]{label=\textcolor{blue}{\textbullet}}

\begin{coloritemize}
  \item First item
  \item Second item
  \item Last item
\end{coloritemize}

\end{document}

You can further customise the output to use a colour of your choice. Here's one such option:

\documentclass{article}

\usepackage{enumitem,xcolor}

\newlist{coloritemize}{itemize}{1}
\setlist[coloritemize]{label=\textcolor{itemizecolor}{\textbullet}}
\colorlet{itemizecolor}{.}% Default colour for \item in itemizecolor

\setlength{\parindent}{0pt}% Just for this example

\begin{document}

\colorlet{itemizecolor}{red}

\begin{coloritemize}
  \item First item
  \item Second item
  \item Last item
\end{coloritemize}

\colorlet{itemizecolor}{blue}

\begin{coloritemize}
  \item First item
  \item Second item
  \item Last item
\end{coloritemize}

\end{document}