Installing a pay-as-you-go Russian SIM card with unlimited traffic on your smartphone is the easiest way to ensure constant access. These are available at airports and most shopping malls.
Wi-fi is common across Russia and usually access is free (or available for the cost of a cup of coffee). You may have to ask for a password (parol) to get online. Most of the time these days you also input your mobile phone number. Sometimes this will need to be a Russian number (ie one starting with +7); if you don't have one, ask a local if you can use their number.
If you don't have your own wi-fi-enabled device, it's probably easiest to get online in the business centres of hotels or at hostels that have a computer terminal.