Gorno-Altaisk, the capital of the Altai Republic, is the logical jumping-off point for most of Altai's attractions. Be sure to get your visa registered here before heading onwards. Also note that if you plan to do any trekking in the high peaks of southern Altai, you'll need a border permit. If you are heading to Ust-Koksa or beyond (including Tyungur), you will also need a border permit. If you are heading to Mongolia, you will not need a permit as long as you don't venture off the M52 highway (aka the Chuysky Trakt).
Altai Border Zones
The Altai Republic’s border zones with China, Kazakhstan and Mongolia have been under the control of the FSB (formerly the KGB) since 2006. Foreigners are required to submit an application for permission to enter these areas. It affects anyone straying off the Chuysky Trakt between Kosh-Agach and the Mongolian border (you do not need the permit in the town or if sticking to the highway); and anyone travelling further south than, and including, Ust-Koksa.
Applications should be made in Russian and must be submitted by fax to the FSB office in Gorno-Altaisk several weeks before your journey (the entire process generally takes at least two months). The application should include passport details (everything, including where and when issued and expiry date), planned route, purpose of journey and home address. When the permit is ready, you must swing through Gorno-Altaisk to pick it up before travelling onward.
Needless to say, this is infinitely easier if you use the services of a travel agency. Travel agencies outside of Western Siberia are unlikely to be able to do this for you. Two local, English-speaking, companies who can help with this are Altair-Tur in Novosibirsk and K2 Adventures in Omsk.
The likely penalty for travelling in a border zone without a permit is a stiff fine and expulsion from the country.