The most convenient way to get to the Altai area is to take a plane
from Moscow to Barnaul (from $530 return, 4 hours flight) and then
take a bus or personal mini-bus to Gorno-Altaisk.
cheapest way is to take a train from Moscow to Novosibirsk (from
$270 one way), then take a train to Biysk (through Barnaul - about
370 km and $50 one way), then take a car or a bus (cheap) to Gorno-Altaisk
and further - take local buses to the mountains along Chuisky Trakt
(the auto-road that goes through the Altai to Mongolia).