Escape "> and <"

A quick solution would be to use inline math mode: $<<$``Welcome''$>>$

edit: if you want to preserve formatting, have you considered \textgreater and \textless ?

I don't think the less-than and greater-than signs are that pretty here. You might try instead:

• $\langle\langle$``Welcome''$\rangle\rangle$ for angle brackets that are narrower and taller than the mathematical relations.
• \guillemotleft``Welcome''\guillemotright for french quotation marks. Make sure you also have \usepackage[T1]{fontenc} in the preamble.

To be honest I don't think any of those look that pretty, but it is what you asked for. :-D

This may be what you want:

\documentclass{article}
\usepackage[T1]{fontenc}

\begin{document}

\guillemotleft``Welcome''\guillemotright

\end{document}

---

For code listing, you can use listings package:

\documentclass{article}
\usepackage[T1]{fontenc}

\usepackage{listings}
\lstset{
  basicstyle=\ttfamily,
  columns=fixed,
  literate={<<}{\guillemotleft}1 {>>}{\guillemotright}1
}

\begin{document}

\lstinline!<<"Welcome">>!

\end{document}

You can modify the literate as you wish.