How can I sum two values and store the result in other variable?

In regular LaTeX, the calc package allows for easy manipulation of length arithmetic:

\documentclass{article}
\usepackage{calc}% http://ctan.org/pkg/calc
\newlength{\mylength}
\begin{document}
\setlength{\mylength}{\textwidth}%
\noindent\rule{\mylength}{20pt}

\bigskip
\setlength{\mylength}{\textwidth-1cm}%
\noindent\rule{\mylength}{20pt}

\bigskip
\setlength{\mylength}{\textwidth-80pt+5mm-1bp}%
\noindent\rule{\mylength}{20pt}
\end{document}

The above deals with lengths. For basic arithmetic using numbers, the fp package. Here is an example using infix notation (Reverse Polish Notation/RPN is also possible via \FPupn):

\documentclass{article}
\usepackage[nomessages]{fp}% http://ctan.org/pkg/fp
\begin{document}
The following arithmetic is easy:
\begin{itemize}
  \item \FPeval{\result}{clip(5+6)}%
    $5+6=\result$
  \item \FPeval{\result}{round(2+3/5*pi,5)}%
    $2+3/5\times\pi=\result$
\end{itemize}
\end{document}

In classical Knuth TeX,

\newdimen\len
\len=\hsize
\advance\len by -1cm

\newcount\cnt
\cnt=1
\advance\cnt by 1

---

eTeX,

\newdimen\len
\len=\dimexpr\hsize-1cm\relax

\newcount\cnt
\cnt=\numexpr1+1\relax

---

LaTeX with calc,

\usepackage{calc}

\newlength\len
\setlength{\textwidth+1cm}

\newcounter{cnt}
\setcounter{cnt}{1+1}

---

LaTeX2e with expl3 (LaTeX3),

\usepackage{expl3}
\ExplSyntaxOn
\dim_new:N \l_len_dim
\dim_set:Nn \l_len_dim {\textwidth + 1cm}

\int_new:N \l_cnt_int
\int_set:Nn \l_cnt_int {1+1}
\ExplSyntaxOff

Since LuaTeX is available, forget all that complicated stuff and do something like:

\directlua{
a = 0
a = a + 1

tex.print(a)
}