Possibly the minimal setting for short text pieces. It provides no hyphenation rules though. (source Univ Berkeley)
- Load the Cyrillic font encoding
OT2 (not T2A,T2B or T2C) and the Latin one T1.
- For the Russian text change locally the encoding to
OT2.
Example,
\documentclass{article}
%VERY IMPORTANT, the principal enconding of the document must be
%the LAST option. In this case T1 (latin encoding).
\usepackage[OT2,T1]{fontenc}
%This command changes the input locally into OT2
\newcommand\textcyr[1]{{\fontencoding{OT2}\selectfont #1}}
\begin{document}
I begin with English, \textcyr{а потом перехожу на русский}.
Very practical!
\end{document}

Note: Remember, in case you need to hyphenate use \-, e.g., \textcyr{а по\-том пе\-ре\-хожу на русский}
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I am aware that OT2 is a so-called obsolete encoding, but, precisely because of that, it is the most lightweight encoding for Russian. Moreover, it is a TeX encoding, hence not only restricted to LaTeX :).
I, personally, had the problem that both the encoding T2A and the package babel collided with two of my custom defined commands, namely, \C and \G. I wonder whether it is good practice to create such short names for the commands in packages that so many people around the world will be using :-\.
Anyway, I use those commands everywhere in a very long document with a lot of included files, so I did not want to spent hours in "Search and Replace".
Thank God, that the encoding OT2 does not collide with any of my commands.
.auxfile when you do changes like that, it will be recreated automatically. – egreg May 04 '13 at 13:13