You must have an invitation to get the visa. The easiest way for a first time traveler is to go to a language school. It will cost you the same as one of those visa services but you will get a lot more (visa + Russian lessons) for your money. They also do all of the registration and save you a lot of hassel and wasted time. DO NOT attempt to go to OVIR and register by yourself. You will be frustrated, angry and never get registered ( which can create a world of problems). DO stay at the school's dorm or apartment housing. There is a new law in Russia that you MUST have your landlord sign you in at a totally different office before you can register. If you go to a school you don't have to deal with any of these problems. You just relax and have fun. The school I go to is inexpensive and even has work/study options.
Another option is to go through a travel agency and get a 30 day tourist visa. But you can only stay at and be registered by a hotel. This will cost a lot more than a school and will be a lot less fun. A school is a great place to make friends and have a much better experience.