A year ago I visited Moscow for 5 days, had gps in my mobile phone and found one cache. Next time I will take my Garmin for sure.
In Moscow there is enough caches on geocaching.com site ; but outside almost nothing. I understand there is different site because of language, and with my very small skills I succeeded to find cache coordinates from geocaching.su site. They seems to let see coordinates without creating an account. Or is it so you find more caches if logging in to site ? Is it free of charge ? I also loaded a gpx file - but my software did not read it. I guess because of language again. But that´s small problem - the few caches on the way is easy to put in manually. Anyway, this gives motivation to learn russian so it´s not so bad. Even one traditional cache feels like big mystery to me
You don't need to make an account on geocaching.ru to see all caches. But if you need to log your visit of course you need to make an account (it's free of charge).
It's a little complicated to work with russian geosite as you need to know Russian but I'll try to help in some cases.
There is a map with all caches. For example, Moscow and Moscow region - http://www.geocaching.su/?pn=103.
Yellow dots on the map with letters T, П and B shows types of caches. T is traditional type. П is Multi-Cache or Puzzle Cache. And B is Virtual one. Well it's kind hard to find П or B type caches unless you speak Russian as you need to read carefully cache description. But for T type you don't really need to know Russian.
Well, let's pay attention to T (traditional) type of caches. Let's check http://www.geocaching.su/?pn=101&cid=5393. On the left site of the page you could find coordinates as well as some other information about the cache. The coordinates are WGS84. Bottom of rating stars you could see some red links. If you need to export coordinates you need to click on "ЭКСПОРТ ТОЧКИ" link. A new window showed up. Choose GPX if you need it for your Garmin device. But you need to choose "перевести в транслит" option as well or you'll get an error on your Garmin device if you don't have Russian symbols.
If you need help with geocaching.su just let me know and I'll make a short translation of description of traditional caches.