For now mid of 2015 year some new suburban trains in Moscow region have access to Free Wi-Fi internet.
Signs showing that Wi-Fi is available usually located on doors and walls in tambour . New trains look like on picture below they are mostly white colored. On older trains do not expect Wi-Fi 😦
New suburban train in Moscow region
I didn’t exactly write down name of Wi-Fi network but name contains cppk letters.
Ones you connected to Wi-Fi network, open a browser and you will be redirected to access page.
Because it is a test service process of registration is very simple. To say really there is no such a process. You just need to check “I agree (Я согласен)” and then tap on Authorize(Авторизоваться). This is shown on picture below.
Authorization page, Wi-Fi in suburban train in Moscow region
Trip Russia 