If you make the first page using \maketitle this may execute \thispagestyle{plain}. You may either redefine page style plain (see the fancyhdr manual for this), or simply add \thispagestyle{fancy} after \maketitle:
\documentclass{article}
\usepackage{fancyhdr,lastpage}
\pagestyle{fancy}
\cfoot{Page \thepage{} of \pageref{LastPage}}
\begin{document}
\title{Title}
\author{Me}
\maketitle
\thispagestyle{fancy}
\end{document}
BTW: If you're using more versatile KOMA-Script class scrartcl instead of standard class article. You may redefine the page style of the page with the in page title head:
\documentclass{scrartcl}
\usepackage{fancyhdr,lastpage}
\pagestyle{fancy}
\cfoot{Page \thepage{} of \pageref{LastPage}}
\renewcommand*{\titlepagestyle}{fancy}
\begin{document}
\title{Title}
\author{Me}
\maketitle
\end{document}
There are more of those page style commands at KOMA-Script, e.g., \chapterpagestyle for the page with chapter heading (only scrbook and scrreprt). Have a look at the KOMA-Script manual for all these commands.
\cfoot{Page \thepage\ of \pageref{LastPage}}(note the\after\thepage) -- otherwise a space will be gobbled. – lockstep Dec 03 '11 at 16:43\cfoot{Page \thepage{} of \pageref{LastPage}}. – Schweinebacke Dec 03 '11 at 16:45