The closest airport is Novokuznetsk airport (NOZ) with direkt flights from Moscow (everyday, air companies "Aeroflot" and "S7"). Departure from Moscow in the evening, arrival to NOZ early in the morning next day. On board about 4,5 hrs.
Other airport for your convenient connections: Novosibirsk (OVB), Kemerovo (KEJ) or Krasnoyarsk (KJA).
From Novokuznetsk you can take a public bus (about 4 hrs) or private taxi to reach Sheregesh (165 km).
From Novosibirsk you can take a public bus (about 9 hrs, 600 km) or by train Novosibirsk-Novokuznetsk-Tashtagol and then by bus or taxi (long way). Please ask for advice.
From Krasnoyarsk take a flight to Novokuznetsk (only once a week), then 165 by high way (public bus or taxi).
It's also better to arrive to Novokuznetsk by train (from Western or Eastern direction). There are regular trains from Moscow, St. Petersburg, Kazan, Novosibirsk, Krasnoyarsk and Vladivostok. From Novokuznetsk you can take a public bus (about 4 hrs) or private taxi to reach Sheregesh (165 km).