I understand that in order to get a visa to visit Russia you need an invitation before you can get a visa.
If you don't have family or friends that you are visiting and you aren't part of an organised tour, how can you obtain an invitation?
Well, everything is for sale. There are companies out there that will provide you with an official invitation for set prices. (try Google)
I've bought a (single entry) business invitation some years ago, and everything worked out fine. I've met several travellers (in Russia) who bought tourist visa invitations.
Make sure you leave the country one day ahead of the expiration date as to avoid trouble when, for example, your train is delayed.
I am in Irkutsk now, and used visatorussia - they were quick and easy to use. Be aware you can only get a visa in your country of residency or citizenship.
I've had to get invites twice for Russia, and it's a fairly simple but weighty process that takes time - you definitely can't rush it - so start as early as possible, but be aware that you can't get the invite until 45 days before you enter the country, just to make things tricky ;)
Hotels can sponsor you if you spend the first nights with them.
How to obtain the visa to Russia depends at least on your nationality.
As a German in Germany, between 2008 and 2010, I obtained three visas via Pul Express GmbH, an agency in Berlin. Pul Express offers all kinds of visas, including multiple entrance business visas valid over the course of one year. They take care of all the invitations and the paperwork. Only you have to fill out an application form, but I don't remember that as being particularly painful.
If you are in a hurry, they can get a visa for you in less than three days.