Text at the right of icon

My kind welcome to TeX.SE. A starting point could be to use fontawesome package without images.

EDIT: The best answer is that the very good user @Mensch. I don't know how long the modified code will last but I hope I've responded to some parts of your comment. The code to me compiles correctly without errors. Here I have used a best answer of @egreg at this link: Justification inside Substack

enter image description here

\documentclass[a4paper,12pt]{article}
\usepackage{fontawesome,mathtools}
\usepackage{xcolor}

%%%%%%%%%%%%%%%%%%%%%%%% @egreg macro %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
\newenvironment{tsubarray}[1]{%
  \vcenter\bgroup
  \Let@ \restore@math@cr \default@tag
  \baselineskip\fontdimen10 \scriptfont\tw@
  \advance\baselineskip\fontdimen12 \scriptfont\tw@
  \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
  \lineskiplimit\lineskip
  \check@mathfonts
  \ialign\bgroup\ifx c#1\hfil\fi
    \normalfont\fontsize\sf@size\z@\selectfont\ignorespaces##\unskip\hfil\crcr
}{%
  \crcr\egroup\egroup
}
\makeatother
\newcommand{\tsub}[1]{\begin{tsubarray}{l}#1\end{tsubarray}}



\begin{document}
\textcolor{red}{\Large\faMapMarker}$\tsub{first row \\ second row}$
\vskip .5cm
\textcolor{red}{\LARGE\faMapMarker}$\tsub{first row \\ second row}$
\vskip .5cm
\textcolor{red}{\Huge\faMapMarker}$\tsub{first row \\ second row}$
\vskip .5cm
\textcolor{red}{\Large\faMapMarker}$\tsub{\Large first row \\ \Large second row}$
\vskip .5cm
\textcolor{red}{\LARGE\faMapMarker}$\tsub{\Large first row \\ \LARGE second row}$
\vskip .5cm
\textcolor{red}{\Huge\faMapMarker}$\tsub{\Huge first row \\ \Huge second row}$


\end{document}

The main issue here is the very poor code of class friggeri-cv. The environment aside is build with package textpos and the environment textblock, which does not allow to nest other environments inside it ...

So if you insist in using this class you have to work around this culprit.

To get the several symbols in front of the data you can use package fontawesome. Then -- for example -- can you use \faEnvelopeO for a symbol for the email address. Because the place is very small for symbol and address it could be you have manually to add an blank to allow LaTeX a line breaking where the blank is:

\faEnvelopeO ~myemail@gmail. com
%                          ^^^

Otherwise you can use that symbols in command \section{\faInbox} too.

So ou can try the following code for your personal data:

\begin{aside}
%
\section{contact}
123 Broadway
City, State 050022
Country
~
\faPhone~+0 (000) 111 1111
{\small\faFax}~+0 (000) 111 1112
~
[email protected]
myweb.wordpress.com
~
\faEnvelopeO ~myemail@gmail. com
\faInbox ~myweb.wordpress.com
\faGithub ~\href{https://github.com/neoben}{github.com/neoben}
\end{aside}

With the following complete code (I used class cv-style which uses other fonts installed on my system, please copy my code to mwe.tex and change then class to friggeri-cv and test the code)

\documentclass{cv-style} % friggeri-cv

\usepackage{fontawesome} % <============================================
\geometry{%
  showframe, % <================ only to visualize typing area & margins
}


\begin{document}

\header{John}{Doe}
\begin{aside}
%
\section{contact}
123 Broadway
City, State 050022
Country
~
\faPhone~+0 (000) 111 1111 % <==========================================
{\small\faFax}~+0 (000) 111 1112
~
[email protected]
myweb.wordpress.com
~
\faEnvelopeO ~myemail@gmail. com % <====================================

\faInbox ~myweb.wordpress.com
\faGithub ~\href{https://github.com/neoben}{github.com/neoben}
\end{aside}
%
\section{skills}
  \vspace{-0.2cm}

Skill 1, skill 2, skill 3, skill 4, skill 5.

\section{education}
\begin{entrylist}
%------------------------------------------------
\entry
{2010--2011}
{M.Sc. {\normalfont in Economics [Grade]}}
{University}
{\vspace{-0.3cm}}
%------------------------------------------------
\entry
{2004--2009}
{B.Eng. {\normalfont in Engineering Management [Grade]}}
{University}
{(Emphasis in ...)}
%------------------------------------------------
\end{entrylist}


\end{document}

results in the following pdf:

resulting ugly pdf

The line breaks do not look very good and I would use a longer first column to get the email address fitting in one line. You can use the code I already showed in my answer (please see the added redefinition of commands in preamble and the added option left=7cm for package geometry, see also deleted blank in email address):

\documentclass{cv-style} % friggeri-cv

\usepackage{fontawesome} % <============================================
\geometry{%
  showframe, % <================ only to visualize typing area & margins
  left=7cm   % <============================= bigger space on right side 5.6cm --> 7cm
}

\renewenvironment{aside}{% <============================================
  \let\oldsection\section
  \renewcommand{\section}[1]{
    \par\vspace{\baselineskip}{\Large\headingfont\color{headercolor} ##1}
  }
  \begin{textblock}{5}(1, 1.87) % <=================================== 3.6 -> 5
  \begin{flushright}
  \obeycr
}{%
  \restorecr
  \end{flushright}
  \end{textblock}
  \let\section\oldsection
}

\renewcommand{\entry}[4]{% <============================================
  #1&\parbox[t]{10.8cm}{% <============================================= 12.8cm --> 10.8cm
    \textbf{#2}%
    \hfill%
    {\footnotesize\addfontfeature{Color=lightgray} #3}\\%
    #4\vspace{\parsep}%
  }\\}


\begin{document}

\header{John}{Doe}
\begin{aside}
%
\section{contact}
123 Broadway
City, State 050022
Country
~
\faPhone~+0 (000) 111 1111 % <==========================================
{\small\faFax}~+0 (000) 111 1112
~
[email protected]
myweb.wordpress.com
~
\faEnvelopeO [email protected] % <====================================!

\faInbox ~myweb.wordpress.com
\faGithub ~\href{https://github.com/neoben}{github.com/neoben}
\end{aside}
%
\section{skills}
  \vspace{-0.2cm}

Skill 1, skill 2, skill 3, skill 4, skill 5.

\section{education}
\begin{entrylist}
%------------------------------------------------
\entry
{2010--2011}
{M.Sc. {\normalfont in Economics [Grade]}}
{University}
{\vspace{-0.3cm}}
%------------------------------------------------
\entry
{2004--2009}
{B.Eng. {\normalfont in Engineering Management [Grade]}}
{University}
{(Emphasis in ...)}
%------------------------------------------------
\end{entrylist}


\end{document}

with the new resulting pdf:

resulting pdf

Please see that you need to add a blank line between telephone number and fax to get them on separate lines (red arrow marked 1) and see that I deleted the blank in the email address (red arrow marked 2).

At last you will have to play with the values in the redefined commands to get a good layout depending on your personally informations.

A last remark: have you considered to use another template? You have a big blank space on the left. There are IMHO better layouts for cvs available ...

Update:

I added the code used by @sebastiano in his answer to my mwe to proof if the code is working with friggeri-cv/cv-style. With a little change (see added \hfill to get the text right justified):

\documentclass{cv-style} % friggeri-cv

\usepackage{fontawesome} 
\usepackage{mathtools} % <==============================================
\geometry{%
  showframe, % <================ only to visualize typing area & margins
  left=7cm   
}

\renewenvironment{aside}{% 
  \let\oldsection\section
  \renewcommand{\section}[1]{
    \par\vspace{\baselineskip}{\Large\headingfont\color{headercolor} ##1}
  }
  \begin{textblock}{5}(1, 1.87) % 
  \begin{flushright}
  \obeycr
}{%
  \restorecr
  \end{flushright}
  \end{textblock}
  \let\section\oldsection
}

\renewcommand{\entry}[4]{% 
  #1&\parbox[t]{10.8cm}{% 
    \textbf{#2}%
    \hfill%
    {\footnotesize\addfontfeature{Color=lightgray} #3}\\%
    #4\vspace{\parsep}%
  }\\}

% code of @Sebastiano
  %%%%%%%%%%%%%%%%%%%%%%%% @egreg macro %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
\newenvironment{tsubarray}[1]{%
  \vcenter\bgroup
  \Let@ \restore@math@cr \default@tag
  \baselineskip\fontdimen10 \scriptfont\tw@
  \advance\baselineskip\fontdimen12 \scriptfont\tw@
  \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
  \lineskiplimit\lineskip
  \check@mathfonts
  \ialign\bgroup\ifx c#1\hfil\fi
    \normalfont\fontsize\sf@size\z@\selectfont\ignorespaces##\unskip\hfil\crcr
}{%
  \crcr\egroup\egroup
}
\makeatother
\newcommand{\tsub}[1]{\begin{tsubarray}{l}#1\end{tsubarray}}


\begin{document}

\header{John}{Doe}
\begin{aside}
%
\section{contact}
123 Broadway
City, State 050022
Country
~
\faPhone~+0 (000) 111 1111 

{\small\faFax}~+0 (000) 111 1112
~
[email protected]
myweb.wordpress.com
~
\faEnvelopeO [email protected] 

\faInbox ~myweb.wordpress.com
\faGithub ~\href{https://github.com/neoben}{github.com/neoben}
~
\textcolor{red}{\Large\faEnvelopeO}~$\tsub{\hfill first row \\ second row}$ % <=======================
~
\textcolor{red}{\LARGE\faMapMarker}~$\tsub{\hfill first row \\ second row}$
~
\textcolor{red}{\LARGE\faMapMarker}~$\tsub{Long first row \\\hfill second row}$
\end{aside}
%
\section{skills}
  \vspace{-0.2cm}

Skill 1, skill 2, skill 3, skill 4, skill 5.

\section{education}
\begin{entrylist}
%------------------------------------------------
\entry
{2010--2011}
{M.Sc. {\normalfont in Economics [Grade]}}
{University}
{\vspace{-0.3cm}}
%------------------------------------------------
\entry
{2004--2009}
{B.Eng. {\normalfont in Engineering Management [Grade]}}
{University}
{(Emphasis in ...)}
%------------------------------------------------
\end{entrylist}


\end{document}

you can compile without error message to this result:

result


You could use enumitem package to realize lists. The following approach could be of some interest to you.

enter image description here

\documentclass{article}

\usepackage{mwe} % loads example-images
\usepackage{graphbox}
\usepackage{enumitem}


\begin{document}

\begin{minipage}{.3\textwidth}
  \begin{enumerate}[labelsep=5mm]
  \item[{\includegraphics[align=c,height=1.5em]{example-image-A}}] some text
  \item[{\includegraphics[align=c,height=1.5em]{example-image-B}}] further interesting text
  \item[{\includegraphics[align=c,height=1.5em]{example-image-C}}] another description etc. etc.
  \end{enumerate}
\end{minipage}

\end{document}