You can have your underscores print as such outside math mode and doing their usual business in math mode:
\documentclass{article}
\usepackage[T1]{fontenc}
\catcode`_=12
\begingroup\lccode`~=`_\lowercase{\endgroup\let~\sb}
\begin{document}
Under_score and $a_{1}$.
\end{document}

For the problem of colored URLs, just load hyperref with the relevant option. With \urlstyle you can also change the default font.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{hyperref}
\hypersetup{colorlinks=false}
\catcode`_=12
\begingroup\lccode`~=`_\lowercase{\endgroup\let~\sb}
\begin{document}
Under_score and $a_{1}$, see at
\url{http://tex.stackexchange.com}
\bigskip
\urlstyle{rm}
This URL uses the normal font
\url{http://tex.stackexchange.com}
\end{document}

hyperref, you can go with\url{http://www.example.com}, theurlpackage offers a similar command. – moewe Dec 22 '14 at 10:21_should work without problens in includegraphics – David Carlisle Dec 22 '14 at 10:25\catcode`_=12in the preamble will do, provided you also have\usepackage[T1]{fontenc}– egreg Dec 22 '14 at 10:26\url{http://.....}(url or hyperref) as then line breaking is better handled but if you want_to be a normal character use\catcode`\_=12make sure you have\usepackage[T1]{fontenc}as OT1 fonts do not have the_character. – David Carlisle Dec 22 '14 at 10:27\usepackage[T1]{fontenc}with\catcode`_=12did what I wanted. Maybe hyperref withurl{http://...}is the proper way, but it has obnoxious defaults (bright teal eyesore around every url - why? why?). – reducing activity Dec 22 '14 at 11:10urlis not adding bright teal eyesore around every url likehyperref, but it is also not working out of the box - urls are refusing to respect margins, despite possibility to fit it by adding line break before url). – reducing activity Dec 22 '14 at 11:21hyperrefadds hyperlinks to URLs (these are by default marked with boxes), you can disable this behaviour or modify it (see, here for example), line breaking in URLs is a tricky issue (see for example here) – moewe Dec 22 '14 at 11:25