What's the best way to explode a string into characters, process each character individually, and join them back?

\documentclass{article}
\def\xloop#1{\ifx\relax#1\else\xloopbody{#1}\expandafter\xloop\fi}
\newcommand\explodethenjoin[3]%
  {\def\inter{\def\inter{#3}}%
   \def\xloopbody##1{\inter#2{##1}}%
   \xloop#1\relax
  }
\begin{document}
\explodethenjoin{abcdef}{\fbox}{+}
\end{document}

The macro \xloop iterates over the tokens following it until it encounters \relax. It has been used several times on this site. For each token the macro \xloopbody is executed; by defining it appropriately, the tokens can be processed as needed.

As TeX doesn't really have strings, you do not need any specific commands to split up a token list, it is already a list of distinct tokens. Similarly you don't have to use an explicit loop macro as iterating over a token list is just the normal tex behaviour.

\documentclass{article}

\def\zz#1{\def\zzsep{}\zzz#1\relax}
\def\zzz#1{\ifx\relax#1\else\zzsep\def\zzsep{+}\fbox{#1}\expandafter\zzz\fi}
\begin{document}
\zz{abcdef}
\end{document}

It's a breeze with xparse:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\splitstring}{mmO{}}
 {
  \spraff_string_split:nnn { #1 } { #2 } { #3 }
 }

\seq_new:N \l_spraff_string_in_seq
\seq_new:N \l_spraff_string_out_seq

\cs_new_protected:Nn \spraff_string_split:nnn
 {
  % split the string at 'nothing'
  \seq_set_split:Nnn \l_spraff_string_in_seq { } { #1 }
  % change each item into '#2{<item>}'
  \seq_set_map:NNn \l_spraff_string_out_seq \l_spraff_string_in_seq { #2 { ##1 } }
  % use the sequence with '#3' in between items
  \seq_use:Nn \l_spraff_string_out_seq { #3 }
 }
\ExplSyntaxOff

\NewDocumentCommand{\boxit}{m}{\fbox{\strut#1}}

\begin{document}

\splitstring{abcdef}{\boxit}

\splitstring{abcdef}{\boxit}[${}+{}$]

\end{document}

Beware that the macro used as second argument to \splitstring should be defined with \NewDocumentCommand.