Companies offer "medical cover" which usually mean that you have the right to the same medical attention that Russian citizens have. Avoid getting into Russia hospital, it may be not very clean there, and the service is not as good as in western European countries. If you have a cell phone take it with you and change the card for a Russian card when you arrive.
Be sure that your apartment will have a working phone otherwise you are unable to contact anyone and you are also uncontestable.
Read English papers that are available in Russia online to be aware of the situation. Try The Moscow Times or The Exile or The Russia Journal for general information and for job opportunities.
You should be careful dealing with getting the visa. You need a visa that lets you enter and LEAVE the country. Some organizations give visas just for entry. Only the school authorities can do this and it takes up to three weeks to obtain. So when the teacher is not happy about the working conditions or something else, he cant6 leave the country. Some offer a simple 3 month tourist visa which can be extended, but which often means that a quick trip to the Baltic States is necessary in order to obtain a new visa.
You need a work permit which the company should obtain before you arrive – as a result many teachers end up working illegally. According to Russian statistics, 60% of foreign teachers work in Moscow illegally. This means that when you are stopped by the police (a frequent occurrence) the only way out of the problem caused by having irregularities with your papers is to pay a bribe in cash. Moreover, all contracts must be written in Russian on one side of the page, vertically, and on the opposite side must be a translation into English.
It must be stamped and signed by a notary public and it cannot be e-mailed, it must be an original copy. Visas and residence permits are usually employer specific, which can make it very difficult to change jobs, should you fancy a change. Also Directors and school owners often exert way too much control over their teachers, banning student contact outside the school and such like. Russia, or at least Moscow, is a very good place for a native English speaker to live and teach. After being here for a while it is normally fairly easy to find hourly paid or private work - in fact it is easy to have too much work as there is far greater demand for native teachers than available teachers!