There are two big issues in moving: language and the financial ability to hang in there until you have an income stream.
If you're older and have a pension from a first world country, if you have enough money for the initial move almost certainly you'll be OK as most first world pensions are more than enough to live adequately, if modestly, in mid-sized cities outside of Moscow or St. Pete, which can be very expensive near the centers. Many Westerners who are moving to Russia are retirees, so they don't have the issue of finding gainful employment.
But if you're still in your working years, you'll need to arrange employment and the old standby from years past of teaching English isn't going to cut it. If you have a profession, like programming or engineering, you'll have no problems so long as it isn't one of the non-portable professions like law that tend to be tied to a specific country. Trades, like welding, are no problem although you should plan for some time, six months to a year, of adapting to local practice.
It takes about six months of intensive effort, several hours a day, for an adult to master Russian from zero knowledge to a level where you can live and work comfortably in the country. Arrive with enough money to pull you through that time and just bite the bullet and do it and you'll be glad you did. It's much easier to learn the language here because you're surrounded by it.
Don't make the mistake of moving to Moscow and living in the expat community, which you can do for many years without learning Russian. Some expats in Moscow are like the Brits who retire to Brittany in France, where they can live in a town surrounded by other Brits and all the signs and menus and such will be in English. I made that mistake and only after I moved out to a town a few hundred miles from Moscow where everybody spoke Russian did I finally get around to learning the language.
Even if you don't need to work, make sure to learn the language. That opens the door to so many really cool and interesting cultural experiences and changes everything. I occasionally run into expats in Russia who complain the people are distant and rude to them. When I ask them how well they speak Russian they usually say "I don't". My reply to them is "What did you expect? You don't speak the language and you think they're being distant to you?" The language barrier is a real deal. If you speak their language, Russians are very friendly. If you don't speak their language, they're cautious and distant.
I recommend living in one of the cities with a population from 250,000 to around a million, because those cities are big enough to have all the services and employment opportunities you want but not so big that costs are driven up by huge populations.
For a first trip to Russia you can get a tourist visa or an education visa. If you're coming here to learn the language and will sign up with a language school it's trivially easy to get a one year visa. Once you have that as you get better with the language you can convert to a one to three year temporary resident visa that will also let you work.
Many foreigners here just leave it at that as you can just renew that temporary resident visa every three years. Or, you can get a permanent resident visa, the equivalent of a US "green card". That gives you all the rights of citizens except the right to vote. To get that you have to pass a language test and short test on Russian history and civics. It's no big deal but many foreigners here (including me) don't want to hassle with that and haven't done it, as it isn't really a significant upgrade from temporary resident status. The main thing it gets you is not having to stand in line every three years with all the guest workers from Central Asia to renew the temporary resident visa.
A few foreigners here get Russian citizenship. That was rare in past years because it used to be that you had to renounce any other citizenships but a few years ago they changed that and allowed you to keep your prior citizenship and be a dual citizen. Applying for citizenship also means passing a much more serious Russian language, history, and civics test than required for permanent residency. That's a real hassle for foreigners who speak, read, and understand Russian with perfect transparency but who have highly imperfect grammar.
I've heard that if you are over 65 you don't have to take the test but I don't know if that is true or not. I don't know any expats who have gotten Russian citizenship. Most everybody I know is living here on a temporary resident visa and some have been doing that for many years.
As to how to start, start by reading everything you can online, all the RT articles and documentaries on life in Russia, the various blogs foreigners who moved here publish, and so on. After that, arrange a two or three week tourist trip with a stop in Moscow (of course) and then travel to midsized towns within several hundred miles of Moscow. Close in towns like Vladimir and Yaroslav, or further out like Rostov on the Don, Krasnodar in the South, Volgograd, Saratov, Samara, Kazan, Nizhny Novgorod, and such. For a more exotic experience, go even farther to Ekaterineburg, or Novosibirsk or Irkutsk by Lake Baikal. or even Vladivostok.
That will give you an impression of different towns to help you decide where you might want to live for a year or two, to learn the language. There will be English speakers you'll know as a result of corresponding with possible language schools who you can use to help you rent an apartment where you decide to live while you learn the language.
Making the transition is easier if you are closer to Moscow, but some foreigners just like to jump right in. You also have to really like winter to go out to the Urals or Siberia (which is not a wasteland but is a huge, beautiful wilderness like Alaska or Canada).
Traps to avoid? Don't break the law, like using illegal drugs. Always be polite and respectful to cops. Don't get so drunk you fight with police. Don't get political as you don't know enough to understand the difference between expressing an opinion and starting a fight.
Enjoy!