Suppress Fancy header and footer on first page only.

Put \thispagestyle{empty} at the beginning of your document. This will only affect the first page:

\documentclass{article}

\usepackage{fancyhdr}

\pagestyle{fancy}
\fancyhf{}
\fancyhead[L]{Top Left}
\fancyhead[C]{Top Center}
\fancyhead[R]{Top Right}
\renewcommand{\headrulewidth}{0.4pt}
\fancyfoot[L]{Bottom Left}
\fancyfoot[C]{\thepage}
\fancyfoot[R]{Bottom Right}
\renewcommand{\footrulewidth}{0.4pt}

\begin{document}\thispagestyle{empty}

Lorem ipsum.

\newpage

Lorem ipsum to you, too, brother.

\end{document}

Note that if you use \maketitle, you'll have to put the \thispagestyle{empty} after the \maketitle because \maketitle triggers a \thispagestyle{plain}, which is the standard page style with only the page number at the bottom.

If you want any other page style on your first page, you can put in anything else instead of empty, of course.

Move the preamble for header and footer into the document after the \newpage where you would like it to start.

\documentclass{article}

\usepackage{fancyhdr}

\begin{document}
%\thispagestyle{empty}

Lorem ipsum.

\newpage

\pagestyle{fancy}
\fancyhf{}
\fancyhead[L]{Top Left}
\fancyhead[C]{Top Center}
\fancyhead[R]{Top Right}
\renewcommand{\headrulewidth}{0.4pt}
\fancyfoot[L]{Bottom Left}
\fancyfoot[C]{\thepage}
\fancyfoot[R]{Bottom Right}
\renewcommand{\footrulewidth}{0.4pt}

Lorem ipsum to you, too, brother.

\end{document}