Hi Sergey and welcome to the forum.
We do have quite a vibrant music scene that is mainly based around our Capital Cities, Sydney and Melbourne being the largest and more bands, many starting off by getting gigs in local pubs.
Like all forms of artists, it is quite possible that band members can have other work when they start off as main playing nights out are Friday to Sunday.
If they find they are good, get a good following etc. and have plenty of work they may decide to go full time into music and then recording music and getting media coverage would just increase their exposure and that is probably a preliminary step to having a successful tour which various groups and solo artists do from time to time and they'll play at quite a few larger regional cities as well as Capital Cities.
In addition to Rock, Country Music is very popular and there are quite a few Country or Folk Music festivals about the country as well as Rock Music Festivals and special events like BIG DAY OUT 2010 - Music Festival - Auckland, Gold Coast, Sydney, Melbourne, Adelaide, Perth
So one way or another you could say that there is a good local scene, whether it be the locals enjoying regular bands at their local pubs or even going to see ones that have become more well known.
You usually find that the festivals and special events are quickly sold out.
I do not think we have ever had a Russian group tour here and with the right kind of promotion it could certainly get attention with a " back in the USSR " flavour.
Can't help you as far as labels, agents or promoters as just not my scene but do a google on Music Agents or Promoters and you'll likely find a few online references.
And best thing for you to do is to get together a Video Clip to send to a few agents for the idea of touring.
You might even want to include parts of Asia in a tour and I reckon you would be a great hit in Vietnam and even be able to do some songs in Russian as the Vietnamese learnt Russian as a second language for many years.
Asia could be a good leaping off platform for a tour as it is a cheap region in which to travel and maybe you even get some Asian female chorus singers and/or a lead singer which could add to stage presentation and there are many Asians in Australia and it could widen your appeal.
Visa and embassy information for all countries - Projectvisa.com you will find is a useful site when you are ready for looking at various country visa requirements.
Good Luck and if you need some help with planning, let me know.