I can buy an invitation that costs 30€ in the consulate and travel everywhere I want. Is there a better way? I also need a registration for three different cities I want to visit, and it’s going to cost me. Is it ok to have a registration for the whole period in one of the three cities even if I’m not staying there for the whole time?
For the cities you need to name in your application, you don't have to visit them exactly. I wasn't sure where I was travelling, and merely named the big cities - St Petersberg, Moscow etc. They just want a rough itinerary of where you plan to go.
Once you're IN the country, you're required to register in a city if you're there for 3 nights or longer. This is a pain. However, the two processes (naming the cities on your application and registering in the cities) are completely unrelated.
Yes, it will cost you each time to register, but you can reduce this by moving around a lot (but keep track of all your train tickets etc, police can ask you for these at any given time).
In theory you could get registered for say, a week in Moscow and travel elsewhere, but if a policeman checks your registration papers and you're meant to be staying in Moscow the next two nights, and you're in Irkutsk, questions may be asked. It's a risk you'd have to consider.
However, if no police check you, once you exit the country they don't care again, they just check that you've not overstayed your visa and that you still have the appropriate documentation from when you entered.
So long story short, to minimise registration costs, keep moving around, but keep track of all your journey ticket etc. However, moving may cost you more than registration, so it's a case of working out your budget.
Source: I've been to Russia twice, once on a tourist visa, once on a multi-entry business visa (for tourist purposes - the regular visa was too short for me).