In general you need to obtain a Russian visa to get into the country. Please be sure to check the list of 43 countries, which citizens are not required to get a Russian visa.
If your country is not on the list, you will have to find the website of the Russian consulate in your country and get information about the process and particular requirements to obtain the visa (it differs from one country to another). You can apply directly in the consulate or in one of the Russian visa centers in your country.
How to get a Russian visa
Usually it takes from 3 to 10 working days to get a visa, but if you mention the Caucasus as your destination it can take up to 1 month.
In general, the consulate or Russian visa center will require you to provide an invitation letter (really that is 2 documents: Voucher and Confirmation) issued by a Russian tour operator. If you are going to the Caucasus with us, we’ll send you the invitation. In most consulates they will be satisfied with just a scanned copy of the invitation, but some consulates require the original paper. In this case we can send it to you with an extra fee for delivery.
Other important factors
If you are going to visit Abkhazia (from Russian side) and South Ossetia count your entrances to Russia. It is highly probable that you will need to apply for a double-entry or even multiple-entry visa.
If you come, for example, to South Ossetia having a normal (1 entry) Russian visa you will be stuck there. They will not let you get back into Russia and you will have to apply for another visa and wait for it in South Ossetia.