You can control the appearance of added and deleted text through the commands \setaddedmarkup and \setdeletedmarkup, respectively.
So you can define, for example:
\setaddedmarkup{\colorbox{yellow}{#1}}
\setdeletedmarkup{\colorbox{red}{\sout{#1}}}
With the following code
\documentclass{article}
\usepackage{changes}
\definechangesauthor[name={Philipp}]{P}
\setaddedmarkup{\colorbox{yellow}{#1}}
\setdeletedmarkup{\colorbox{red}{\sout{#1}}}
\begin{document}
This is \added[id=P]{new} text.
This is \deleted[id=P]{bad} text.
This is \replaced[id=P]{nice}{bad} text.
\end{document}
you get

This doesn't obviously support line breaks.
If you don't need particular formatting apart from coloring, you can load the soul package (once loaded by changes), and define, for example:
\setaddedmarkup{\sethlcolor{yellow}\hl{#1}}
\setdeletedmarkup{\sethlcolor{red}\hl{#1}}
MWE
\documentclass{article}
\usepackage{changes}
\usepackage{soul}
\definechangesauthor[name={Philipp}]{P}
\setaddedmarkup{\sethlcolor{yellow}\hl{#1}}
\setdeletedmarkup{\sethlcolor{red}\hl{#1}}
\begin{document}
This is \added[id=P]{new} text.
This is \deleted[id=P]{bad} text.
This is \replaced[id=P]{nice}{bad} text.
\end{document}
Output:

Otherwise, have a look at this thread for more highlighting possibilities: Cool Text Highlighting in LaTeX